PHYSICS UNION MATHEMATICS Physics II - · PDF filePHYSICS UNION MATHEMATICS Physics II ... Ranking Task Exercises in Physics ... Think about how the motion of the ball and the marks

P H Y S I C S U N I O N M A T H E M A T I C S

Physics II Kinematics

Supported by the National Science

Foundation (DRL-0733140).

2 PUM | Kinematics | © Copyright 2013, June, Rutgers, The State University of New Jersey.

PUM Physics II

Kinematics

Most of the module activities were adapted from:

A. Van Heuvelen and E. Etkina, Active Learning Guide,

Addison Wesley, San Francisco, 2006.

Used with permission.

Some activities area based on FMCE (Thornton and Sokoloff), on

Ranking Task Exercises in Physics (O’Kuma, Maloney, and

Hieggelke), and on the work of D. Schwartz.

Contributions of: E. Etkina, T. Bartiromo, S. Brahmia, J. Chia, C. D’Amato, J. Flakker, J.

Finley, H. Lopez, R. Newman, J. Santonacita, E. Siebenmann, R. Therkorn, K. Thomas, M.

Trinh.

This material is based upon work supported by the National Science Foundation under

Grant DRL-0733140. Any opinions, findings and conclusions or recommendations

expressed in this material are those of the authors and do not necessarily reflect the views of

the National Science Foundation (NSF).

PUM | Kinematics |

© Copyright 2013, June, Rutgers, The State University of New Jersey.

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Table of Contents

LESSON 1: ESSENTIALS TO DESCRIBING MOTION 4

LESSON 2: DESCRIBING MOTION 9

LESSON 3: REASONING IN KINEMATICS: INVENTING AN INDEX 14

LESSON 3A: HOW FAST ARE YOU MOVING? 19

LESSON 4: THE MOVING MAN 26

LESSON 5: HOW MANY VELOCITIES CAN AN OBJECT HAVE? 32

LESSON 6: USING THE LANGUAGE OF MATHEMATICS WHILE DOING PHYSICS 37

LESSON 7: USING VELOCITY-VERSUS-TIME GRAPHS TO FIND DISPLACEMENT 41

LESSON 8: LAB: WILL THE CARS EVER MEET? 46

LESSON 9: MOTION DIAGRAMS 52

LESSON 10: FREE FALLING? 57

LESSON 10B*: THE TRUTH BEHIND GRAPHIC REPRESENTATIONS 58

LESSON 11: MOTION OF A FALLING OBJECT 60

LESSON 12: POSITION OF AN ACCELERATING OBJECT AS A FUNCTION OF TIME 71

LESSON 13: EXPERIMENTAL DESIGN 77

LESSON 14: DETAILS OF THE THROW 79

LESSON 15: PUTTING IT ALL TOGETHER 82

LESSON 16: PRACTICE 8889

4 PUM | Kinematics | Lesson 1: Essentials to Describing Motion © Copyright 2013, June, Rutgers, The State University of New Jersey.

Lesson 1: Essentials to Describing Motion

1.1 Observe and Find a Pattern

In this activity, a tube “telescope” is used to follow an object. During the experiments, the

observer always keeps the object in sight through the telescope. Consider the following

situations:

a) In the first experiment, the teacher holds an object and is standing still. You are the

observer; take note of the initial direction that the telescope points when you see the object

through it. This is the original orientation of the telescope. Next, the teacher travels from

right to left.

- What happens to your telescope as you follow the object?

- Draw a rough sketch to show what happens to the orientation of the telescope.

b) For the next experiment, carefully observe your teacher. The teacher holds the ball the

same way. This time, the teacher is the observer and looks at the object through the

telescope while traveling from right to left.

- What happens to the orientation of the teacher’s telescope during the experiment?

c) Based on your experiences in these two observational experiments:

- Can you say whether the ball was moving during the two experiments? Explain

your answer. (Hint: Compare what the different observers had to do to keep seeing

the ball through their telescopes.)

- In general, how do you know whether something is moving or not?

- Name two observers who see you moving right now. How do you know?

- Name two observers who see you stationary right now. How do you know?

d) What if….

- Chris watches a car through his scope. He is turning his scope to the left. In what

direction is the car moving?

- Jodi watches the same car but she is not moving the scope. How can this be? What

might happen to the size of the car as Jodi watches it?


Consider the following situation.

• A blue car moves along a street with two passengers. One sits in the front passenger seat of

the car and the other passenger sits in the back seat.

PUM | Kinematics | Lesson 1: Essentials to Describing Motion


5

• A red car moves in the same direction and is passing the blue car.

• A green car moving faster than the blue car, is directly behind the blue car

• There is a sidewalk along the road the cars are traveling and a pedestrian is standing on the

sidewalk.

Choose four students from your class to play the roles of the four people and recreate

the situation.

Describe the movement of the front passenger in the blue car as seen by each of the

following observers:

Observer Describe what she/he sees

The person sitting in the backseat of

the blue car.

The pedestrian standing on the

sidewalk as the blue car passes.

The driver of the red car moving in the

same direction and passing the blue

car.

A passenger in the green car

Review your descriptions and answer the questions that follow.

a) Imagine you are the backseat passenger in the blue car, how would you observe the other

four observers? Explain.

b) Imagine you are the pedestrian in the street, how would you observe the other four

observers? Explain.

c) Based on your answers above, explain what it means when someone says an object is

“moving”.

d) Consider the phrase “motion is relative”. Use your idea of what it means to move to explain

the meaning of this statement.

1.3 Represent and Reason

Examine the map below. Your friend is visiting Washington DC and is staying at George

Washington University. Your friend must walk across town to get to the Smithsonian

Institution.


a) Provide simple directions to your friend so she can get to the Smithsonian by 8:30 am.

b) What was the object of reference that you selected to provide directions?

c) How did the map legend help you to give essential directions?

d) What were the assumptions you made when you provided directions?

Need Some Help?

Assumptions are issues we take for granted in an experiment. They can help us explain

why the results weren’t exactly as we expected or could be something that wasn’t taken into

account when we designed the experiment. Despite the outcome of the experiment, it’s

important to consider the assumptions that may have been made.

Assumptions are factors that could affect our results, but have not been included in our

calculations or reasoning.

e) How did this affect the time you tell your friend to leave?

Did You Know?

Motion: An object is in motion with respect to another object (reference object) if, as time

progresses, its position is changing relative to the reference object.

Reference frame: A reference frame includes three essential components:

PUM | Kinematics | Lesson 1: Essentials to Describing Motion


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1. An object of reference, which is a real object in the physical world.

2. A clearly defined coordinate system. The coordinate system includes labels for the

direction of the axis, such as north, south, east, west, left, right, up down, or positive and

negative. The unit scale for measuring distances is also identified. A point on the coordinate

system, usually the 0 point, is attached to the object of reference.

3. A zero clock reading that serves as a reference for future clock readings.

1.4 Reason

Meagan and Beccy are sitting in a train, which is carrying them east toward New York City.

They see Ryan walking down the aisle toward the rear of the train. He is texting on his

phone and does not see that he's about to bump into Andrew, who is standing in the aisle.

Meagan says "Ryan must moving west, because he is about to bump into Andrew."

Beccy says "That can't be true, because if Ryan was moving west he would never get to

New York City."

Meagan says "But he can't be moving east or else he would not be about to bump into

Andrew!"

a) Act out the situation.

b) Do you think Meagan or Beccy is correct? Is either one wrong? Explain how you

evaluate their reasoning.

c) Explain carefully what is REALLY correct about the motion of Ryan.

Homework

1.5 Relate

Describe three situations from your life that are relevant to your idea of relative motion and

reference frames.

1.6 Explain

a) Using the idea of a reference frame provide directions to get to school from your

house 15 minutes before homeroom begins.

b) When we say the Sun is rising and the Sun is setting, who is the observer? Where

should the observer be to see that the Sun does not move around the Earth every 24

hours?

c) Someone in your class says the “Sun does not move,” then why do we have days

and nights?


1.7 Observe and Explain

a) Use what you have learned so far about describing motion to explain the sequence of

photographs using two different reference frames.

Photo

1

Photo

2

b) Imagine that you were an observer somewhere in the picture above, where should

you be located to see:

• the student moving? the student not moving?

• the street sign moving?

• the police car moving right? the police car moving left?

• the bicycle moving? the bicycle not moving?

c) From photos 2 and 3 above, where would an observer have to be located so that she

sees the pickup truck moving backwards? Explain why this is the case.

Reflect: What did you learn during this lesson? Can you explain to

your friends or relatives what the words “motion is relative” mean?

What is a reference frame and why is it important to know about it?

PUM | Kinematics | Lesson 2: Describing Motion


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Lesson 2: Describing Motion

2.1 Observe and represent

Scientists communicate with each other using pictures, words, mathematical relations,

graphs, and many other representations. In this observational experiment, you are going to

learn how to represent motion in various ways. Before you begin, place a ball on the metal

track. Practice rolling the ball along the track with one of your teammates.

a) Describe the motion of the ball as best you can.

b) With your teammates, think of a way to keep the clock reading and mark the location of the

ball for each second as it rolls. You can use any marking method you think is appropriate.

Describe your method and practice until you are comfortable.

Here’s an Idea! - To practice your counting look at a second hand on a stopwatch while

tapping on a desk or shouting a word.

If you yell out, make sure that when you shout your counts for each second, they are brief;

for example “yes”.

c) Once your team decides on a procedure, perform the experiment.

d) Designate an origin and record each clock reading and position for each mark: be sure to

include the axis and designate directions.

e) Revise your description in part (a).

f) Explain how you can use the marks to describe the motion of the ball

2.2 Hypothesize

Think about how the motion of the ball and the marks are related in general. Propose your

hypothesis here.

2.3 Test Your Idea

Imagine that you now let the ball roll down the track that is tilted with respect to he

horizontal. Use the hypothesis in 2.2 to predict the relative spacing of the marks for the ball

rolling down the incline.

10 PUM | Kinematics | Lesson 2: Describing Motion © Copyright 2013, June, Rutgers, The State University of New Jersey.

Rubric to self-assess your prediction

Scientific

Ability

Missing An attempt Needs some

improvement

Acceptable

Is able to

distinguish

between a

hypothesis

and a

prediction.

No prediction is

made. The

experiment is not

treated as a

testing

experiment.

A prediction is made

but it is identical to

the hypothesis.


and is distinct from

the hypothesis but

does not describe the

outcome of the

designed experiment.

A prediction is

made, is distinct

from the

hypothesis, and

describes the

outcome of the

designed

experiment.

Perform the experiment and compare your prediction to the outcome of the testing

experiment? What does this tell you about your hypothesis?

Did You Know?

One representation of motion is called a dot diagram. To make a dot diagram mark

locations of a moving object at equal time intervals.

2.4 Reason

a) Imagine that you could observe an airplane flying from New York to Los Angeles.

The length of the airplane is 32 m. The length of the trip is about 4000 km or

4000000 m. The average speed of the plane is 500 km/h. You need to calculate the

time that it takes the airplane to fly from NY to LA. Do you need to take the size of

the plane into account? Explain.

b) While traveling to the gate the plane turns left. In order to describe how much a

passenger moved with respect to the ground, do you need to take the size of the

airplane into account? Explain.

c) A passenger is sitting on the plane. The moves plane moves forward 100 m. Do you

need to take the size of the airplane into account in order to determine how far the

passenger moved? Explain.

d) The same plane parks at the gate. You need to calculate how close the gates can be

to each other for the airplanes to park safely. Do you need to take the size of the

plane into account? Explain.

e) When is it important to take the size (or dimensions) of the plane into account?

Did you know?

Point-like object: When you do not need to take the size of the object into account to

solve a problem, you can represent this object as a point. This point will have all

properties of the object except its size and parts. This simplified object is called a

PUM | Kinematics | Lesson 2: Describing Motion


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point-like object. A point-like object is a simplified version of a real object. We consider

real objects to be tiny point-like objects under two circumstances: (a) when all their parts

move in same way, or (b) when the objects are much smaller than the dimensions of the

process described in the problem. [or] when the size and shape of the object is not

particularly important to the situation. The same object can be modeled as a point like

object in some situations and not in others.

f) How did a point like object play a role in the experiments at the beginning of lesson 2?

Homework

2.5 Represent and reason

Below there are images of snail crawling along a table next to a ruler.

Clock reading = 0 s

Clock reading = 1 s

Clock reading = 2 s

Clock reading = 3 s

Clock reading = 4 s

Photographs like the ones seen above can be used to study the motion of objects.

a) Determine the reference frame for these photographs.

b) Record the position of the snail for each clock reading. What assumption(s) did you

make?

12 PUM | Kinematics | Lesson 2: Describing Motion © Copyright 2013, June, Rutgers, The State University of New Jersey.

c) Create a dot diagram for the motion of the snail.

d) How would your picture look different if the motion happened over 8 seconds rather

than 4 sec?


a) Meg created a dot diagram for a bug she found on the ground. Use the dot diagram to

describe the motion of the bug. Is it moving at a constant pace, is it speeding up or slowing

down? Explain.

What

would have

to be true about an observer that draws the same dot diagram as Meg?

b) The dot diagrams for the two bicyclists are show below. When were they at the same

location at the same time (circle)?


Imagine you are taking a road trip to Cape May, NJ (the Southern most part of NJ) and you

program the directions into your GPS. The GPS says that it will take you 2 hours to travel.

When you leave, it is 8:00 am but by the time you arrive it is 10:45 am.

What assumptions did the GPS program make about the trip to Cape May?

What assumptions may have been incorrect? Explain.

If you did arrive at Cape May at 10:05 am, were the assumptions the GPS system made

fair? Explain.

Reflect: What did you learn during this lesson? Can you explain to your

friends or relatives why a dot diagram is useful and why we care about point-

like models of objects?

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●

Bike 1 ● ● ● ● ● ● ●

Marks on the road I I I I I I I I

20m 60m 100 m 140m

Bike 2 ● ● ● ● ● ● ● ●

PUM | Kinematics | Lesson 3: Reasoning in Kinematics: Inventing an Index


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Lesson 3: Reasoning in Kinematics: Inventing an Index

An index is a number that helps people compare things.

• Miles per gallon is an index of how well a car uses gas.

• Batting average is an index of how well a baseball player hits.

• Grades are an index of how well students perform on a test.

We want you to invent a procedure for computing an index that helps make comparisons.

RK.1 The Popping Index

Three companies make popcorn.

They use different types of corn so the popping is fast or slow.

Invent a procedure for computing a “popping index” to let consumers know how fast each brand

pops.

Rules for the Index

1. The same brand of popcorn pops at the same speed. So a

brand of popcorn only gets a single popping index.

2. You have to use the exact same procedure for each brand to

find its index.

3. A big index value should mean that the popcorn pops faster.

A small index number should mean that the popcorn pops

slower.

14 PUM | Kinematics | Lesson 3: Reasoning in Kinematics: Inventing an Index © Copyright 2013, June, Rutgers, The State University of New Jersey.



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RK2. Fastness Index

Let’s look at another kind of index. Your task this time is to come up with a fastness index

for cars with dripping oil. You will see a bunch of cars, and you need to come up with one

number to stand for each car’s fastness. There is no watch or clock to tell you how long

each car has been going. However, all the cars drip oil once a second. (They are not very

good cars!)

You can look at the oil drops to help figure out how long a car has been traveling.

This task is a little harder than before.

• A company always makes its cars go the same fastness.

• We will not tell you how many companies there are.

• You have to decide which cars are from the same company. They may look

different!

To review:

(1) Make a fastness index for each car.

(2) Decide how many companies there are.

(3) To show the cars that are from the same company, draw a line that connects the

cars.

Start

C

A

D

E

F

BStart

Start

Start

Start

Start

Oil

16 PUM | Kinematics | Lesson 3: Reasoning in Kinematics: Inventing an Index © Copyright 2013, June, Rutgers, The State University of New Jersey.

RK.3 Reason

1. Which popcorn is fastest? Give an explanation why you picked that popcorn.

2. Which cars are fastest? Give an explanation why you picked that car.

3. For each question below, describe the steps that you would take to get an answer:

• A full bowl of popcorn has 60 popped corns in it. How long does it take the

fastest popcorn to fill a bowl of popcorn?

• How long does it take for the fastest car to travel 15 blocks?

• How many popped corns will the fastest popcorn pop in 70 seconds?

• How far will the fastest car travel in 20 seconds?

4. Another company, “Acme,” has an index of 2.5.

a) Let’s say that Acme makes popcorn. Using everyday language, describe the

specific information that the number 2.5 tells about this their popcorn.

b) Make a sketch that explains your answer to part a).

c) Now let’s say that Acme makes cars. Using everyday language, describe the

specific information that the number 2.5 tells about their car.

d) Make a sketch that explains your answer to part c).

5. Explain why you think there were less than 6 car companies, even though there were

six different diagrams describing the car companies.

RK.4 The Steepness Index

Let’s try another kind of index.

As the owner of 2-Die-4 Water Park, you are in charge of buying the slides. Most of your

clients are teenagers, and they like the steepest slides they can find. You want to buy slides that

will attract the most business. You are trying to choose between the slides shown below.

Invent a procedure for computing a “steepness index” so that you can buy the best slides, and

prove to your customers that you have the steepest slides in town.

Rules for the Index

1. Each slide gets one steepness index because it has the same

steepness all the way down.

2. You have to use the exact same procedure for each slide to

find its index.

3. A big index value should mean that the slide is steep. A small

index number should mean that the slide is less steep.



17

Find the steepness index for the following portions of these slides

RK5.Reason

1. Which slide is steepest? Give an explanation why you picked that slide.

2. For each question below, describe the steps that you would take to get an answer:

• In another portion of the steepest slide, it drops down by 14 feet. How far does

the slider move horizontally when she goes down that part of the slide?

• How far down has she dropped when she has moved horizontally 5 feet?

3. Another company, “Acme,” has an index of 0.75.

a) Let’s say that Acme makes slides and 0.75 is the steepness index. Using

everyday language, describe the specific information that the number 0.75 tells

about their slides.

b) Make a sketch that explains your answer to part a).

4. A quality control officer for Hop-On Popcorn counted the number of kernels popped

at several different times. The data are shown below in the table.

Popped

Kernels

Time (sec) a) Make a graph of the popped kernels vs. time.

b) Find the steepness index of the best-fit line. What

information does the steepness of the best-fit line

tell you about Hop-On Popcorn?

c) Which pops faster, the Hop-on Popcorn or HipHop

Popping Corn? Explain.

8 6

12 9

16 12

4 3

d) Sketch a bowl of Hop-On Popcorn after 15 seconds in an air popper.

12 ft

16 ft 18 ft

18 ft

7 ft

5 ft

Rocks Ur Socks Splash Attack

20 ft

24 ft

Super Soaker Tsunami

18 PUM | Kinematics | Lesson 3A: How Fast Are You Moving? © Copyright 2013, June, Rutgers, The State University of New Jersey.

Lesson 3A: How Fast Are You Moving?


Roll out a 10 m tape measure across the classroom floor. The teacher will assign each

group a different starting position and direction for their cars (a fast and slow car). For each

car, you will mark the position along the tape measure at each second. Make sure the cars

travel at two different rates.

a) Discuss with your

teammates and

decide on a method

for keeping time and

marking the location

of the car each

second. When you

are ready, start with

your slow car; record the data in the table below. Repeat the experiment for the fast car and

record the data in the table below. Be sure the position represents the location of the car

with respect to the tape measure on the floor.

b) What are the physical quantities measured in this experiment? What units of measure did

you use?

c) Explain the differences between these two ideas.

Did You Know?

Physical Quantity: A physical quantity is a characteristic of a physical phenomenon that

can be measured. A measuring instrument is used to make a quantitative comparison of this

characteristic and a unit of measure. Examples of physical quantities are your height, the

speed of your car, or the temperature of air or water. If a characteristic does not have a unit,

it is not a physical quantity.

Position x is the location of an object relative to a chosen zero on the coordinate axis.

Time Interval: The time interval is the difference between two clock readings. If we

represent one time reading as t1 and another reading as t2, then the time interval between

those two clock readings is t2 - t1.

Another way of writing this statement is:

t2 − t1 = ∆t

The symbol ∆ is the Greek letter delta and in physics and mathematics it is read as delta t

(∆t) or the change in t. Time can be measured in many different units, such as seconds,

minutes, hours, days, years, and centuries, etc.

Slow Car Fast Car

Clock Reading

t

Position

x

Clock Reading

t Position

x

PUM | Kinematics | Lesson 3A: How Fast Are You Moving?


19

Clock reading: The clock reading or time (t) is the reading on a clock, on a stopwatch, or

on some other time measuring instrument.

Example: You start observing the motion of your car when stop watch shows 3 seconds,

when you finish the watch shows 15 seconds. What is the time interval for your

observations?

For your observations t1=3 seconds

t2= 15 seconds, thus t2 − t1 = ∆t = 15 − 3 = 12 s . Here the physical quantity is ∆t and the

units are seconds.


a) Robin, James, Tara and Joe (at rest with respect to each other) collected data for the

motion of the same car. They each represented the data differently. Examine the

four representations below; select a representation that would best represent the

position of the car as a function of time. Explain.

Robin

Tara


James

Joe

b) Discuss your choice and reasoning with the class.

c) Represent the motion of the cars with a graph (plot the data from each car on the same axes)

using the data collected in activity. The position of the car is recorded on the vertical axis

and the clock reading on the horizontal axis.

Did You Know?

In science, experimenters put time on the horizontal axis when it is the independent

variable.

d) Draw a trend line for each car on the graph you drew in part (c). What information can you

learn about the motion of the car from the graph? Explain.

Need Some Help?

Trend Line: A trend line represents a trend in the data. To draw a trend line, try to draw a

smooth line that passes as close to all data points as possible. The data points do not need to

be on the line.


a) Compare the trend lines of the two cars? How are they different?

b) Find the slopes of the two lines. Explain how you found the slope. What name

could you give to the slope?

c) Explain what it means if the slope is positive or negative.



21

Did You Know?

Velocity of an object moving at constant velocity is the slope of the position versus time

graph and is equal to the change in position of the object divided by the time interval during

which this change in position occurred. When the object is moving at constant velocity this

ratio is the same for any time interval

v =x2 − x1

t2 − t1

=∆x

∆t

where x2 – x1 ( ∆x ) is any change in position during the corresponding time interval t2 – t1

( ∆t ). The unit for velocity is m/s, miles/h, km/h, and so forth.

Positive velocity means that the object is moving in the positive direction; negative velocity

means it is moving in the negative direction.

Speed is the magnitude of velocity, it is always positive.

d) What is the velocity of each car in your experiment? How did you do that?

e) How is velocity related to the concept of index you invented in lesson 3?

f) Write a function x(t) for the fast car and a separate function for the slow car. What role

does the trend line play and what role does the y-intercept play in writing this function?

Need Some Help?

When mathematicians and physicists express patterns mathematically they use functions. A

function is a rule that one uses to find a dependent variable when an independent variable is

known. You may have met functions in a math class. There the independent variable was

labeled x and the dependent variable is labeled y. The function then is y(x). In science and

math class you can actually use any labels as long as you agree on which was the

independent and which is the dependent variable. For the problem below, the independent

is t, and a dependent is x. Example:

This expression can all be written in function notion as x(t) = 50t, however, in physics it is

necessary to include units of measure x(t) = 50(m/s) t or x(t)=50(m/s)t. x(t) is read as “x of

t.”

Examine: Describe the relationship

between the two variables.

The object changes its position by

50 meters each second

Define: Describe the variables used

in the scenario

t = time elapsed

x = position

Represent: Write a mathematical

equation using variables x = 50t

Time (second) Position (meters)

1 50

2 100

3 150


REMEMBER! When you graph a function the independent variable is always placed on

the horizontal and the dependent variable on the vertical axis. Position is on the vertical axis

because position "depends on" clock reading.

Homework

3.4 Practice

a) A car moved from x1 = 20 mi to x2 = 62 mi. Draw a picture with the coordinate axis,

zero point and the locations x1 and x2

, and find ∆x .

b) A car moved from x1 = 120 mi to x2 = 34 mi . Draw a picture with the coordinate axis,


, and find ∆x .

c) A car moved from x1 = −30 mi to x2 = 62 mi. Draw a picture with the coordinate axis,


, and find ∆x .

d) A car moved from x1 = −20 mi to x2 = −78 mi . Draw a picture with the coordinate axis,


, and find ∆x .

e) A car moved from x1 = −62 mi to x2 = −20 mi . Draw a picture with the coordinate axis,


, and find ∆x .

3.5 Practice

In the previous example, the time interval during which the position change occurred ∆t =

1.5hr. Determine the velocity and the speed of the car for each ∆x . What does it mean if

velocity is positive? Negative? To answer – relate to the direction of the x axis.

3.6 Analyze

a) The graph below shows the motion of a football player during 20 seconds. What is the

player’s position at the point shown with the triangle on the graph? Choose the answer that

you think is best.

I) 2.5 yards;

II) 10 yards;

III) 35 yards;

IV) 25 yards.

b) How far did the player travel from the beginning of observations?

I) 2.5 yards;

II) 20 yards;

III) 35 yards;

IV) 25 yards.

c) What happened at the 0 clock reading:



23

I) The player started moving;

II) The player was passing the mark of 45 yards;

III) The player was moving in the negative direction;

IV) both II and III are correct.

d) Which answer best describes the player’s motion at the point indicated by the triangle on

the graph?

I) The player is moving at constant speed: II) The player encountered a dip and is moving

slightly downhill; III) The player is slowing down; IV) The player stopped.


Two objects are moving in the same direction. The speed of one is 5 m/s and the speed of

the other is 10 m/s. When you start observing them, they pass the same location at the same

time.

a) Draw dot diagrams for two objects.

b) Represent their motions with position versus time graphs. Use the same scale for both

objects.

b) Choose from position versus time functions describing their motions a combination

that looks correct to you:

I) x1 = (5 m/s) + t; x2 = (10 m/s) + t;

II) x1 = (5 m/s)t; x2 = (10 m/s)t;

III) x1 = t(5 m/s); x2 = t(10 m/s);

IV) both II and III are correct.

c) How long will it take each object to travel 276 m?


d) How far from each other will they be in 10 seconds? 20 seconds after you start

observing them?


The motion of object A is represented by the function xA

= (5.0 m/s)t ; the motion of object

be is represented by the function xB

= (−3.5 m/s)t .

a) Say everything you can about the motions of those objects. If you need to assume

something, state your assumption clearly.

b) Represent the motions in as many different ways as you can.

3.9 Practice

A train is moving at the speed of 15 m/s. How far will it move in 10 seconds? In 10

minutes? In 10 hours?

3.10 Practice

You are riding a bicycle to your friend’s house. The house is 3 km away. You arrive at the

house in 17 minutes.

a) What was your speed? Write the speed in km/min; in m/s; and in mph. List all

assumptions that you made.

b) Write a function x(t) for your ride. In how many ways can you write this function?

3.11 Practice

Usually, a briskly-walking person can cover 4 miles in an hour. How long will it take this

person to walk 12 miles? 0.3 miles? 4 kilometers? What assumptions did you make?

3.12 Reason

You walk 1.8 miles every 30 min. Use the index approach to calculate in your head how far

you will walk in: (1) 1 hour; (b) 1 hour 30 minutes; (c) 2 hours.

Reflect: What did you learn in this lesson? Can you explain to your

friends how the slope of the position versus time function is related to the

object’s velocity? What does it mean if velocity is positive? What does it

mean if it negative?

PUM | Kinematics | Lesson 4: The Moving Man


25

Lesson 4: The Moving Man

4.1 Hypothesize

An object is moving in the positive direction at constant velocity v. It starts at clock reading

t = 0 sec, at a position x0. How would you write a function that will allow you to find the

position of the object at any time?

4.2 Test Your Idea with Phet Simulations

Go to http://phet.colorado.edu/web-pages/simulations-

base.html and click on the simulation The Moving Man.

Browse to http://tinyurl.com/4lwnlc (or google for

phet moving man) and click Run Now

You should see a screen like the one shown. There is a man

at top of the simulation who can move 10 m in either

direction from the origin. The simulation also includes

axes of position, velocity and acceleration graphs that will

reflect his motion. Since you are not going to use the

acceleration or velocity graph right away, you can close them by clicking on the small

window in the upper right hand corner of each section.

To eliminate the walls, click on “special features,” then click on “free range”.

Use the hypothesized mathematical model in activity 4.1 to predict the position.

Scenario 1: The man’s initial position is 9 m and he is jogging to the left at 2 m/s.

a) Write an expression for the man’s position as a function of time.

b) Create a position vs. time graph for this function.

c) Before you continue with the simulation, check for consistencies between the

written description, function and graph for the man. How do you know they are

consistent?

d) Predict the time when he passes through the position at 0m.

e) Perform the experiment by entering given quantities in the respective simulation

boxes and click Go! Compare your predicted value to the outcome of the testing

experiment. Do they agree or disagree? If they disagree, revise your mathematical

model of the moving man’s motion.

Scenario 2: The man is walking at the speed of 0.75 m/s towards his home. When we start

observing him, he is at the position of 7 m to the left of the origin.


26 PUM | Kinematics | Lesson 4: The Moving Man © Copyright 2013, June, Rutgers, The State University of New Jersey.




consistent?

d) Predict the time when he arrives at the house.

e) Perform the experiment through the simulation. Compare your predicted value to the

outcome of the testing experiment. Do they agree or disagree? If they disagree,

revise your mathematical model of the moving man’s motion.

Scenario 3: When we start observing the man is at the 5 m mark by the house and is

running at the speed of 4.5 m/s towards the tree.





consistent?

d) Predict the time when he has traveled 70 m beyond the tree.




4.3 Test Your Idea

Use your newly modified hypotheses from the previous activity to predict how you’d have

to move so that a motion detector creates position versus time graphs that match the graphs

in the previous activities. Explain how your prediction compares to the outcome.


In this activity you will be acting as the Moving Man or Woman. Below are a number of

position vs. time functions and written descriptions. Here you will have to act out how to

move so that a motion detector creates position versus time graphs match description

provided. Before acting it out, discuss how your motion should match the written and

mathematical descriptions.

a) A person is 7.0 m away when we start observing and walks towards the origin at 0.4

m/s.



27

b) You are 5.0 m away from the origin at the initial clock reading and walk at 1m/s for

4 seconds towards the origin stop for 3 seconds then walk 0.5 m/s for 6 seconds

away from the origin.

c) x(t) = 5m +(– 0.7m/s)t

d) x(t) = 0.8m + (1.2m/s)t

Homework


Examine the dot diagram above. When we start observing the object it is at +7.5 m and

moves in the negative direction of the x-axis.

a) Describe the motion in words.

b) Sketch a position vs. clock reading graph.

c) Write a function for the position as a function of time for the object’s motion.


Thus far, you have represented the motion of a ball with different representations: dot

diagrams, words, pictures, tables, and now a graph. Explain how the different

representations describe the same motion.

a) Use the graph below to describe the motion in words. Pay attention to what

happened at zero clock reading!

b) Write a function for the position as a function of time for the object’s motion.

c) You should notice that the

two physical quantities on

the graph do not have units.

Describe a real life situation

for this motion if the units

were kilometers and seconds.

Describe another situation if

the units were centimeters

and minutes.

d) Draw a picture for each of

the situations you described in part c.

0.0 m +5.0 m +10.0 m -5.0 m -10.0 m

∙ ∙ ∙ ∙ ∙


4.7 Equation Jeopardy

Three situations involving constant velocity are described mathematically below.

1. (−86 m) = v(1.72 s) + (−100 m)

2. x = (−5.7 m/s)(300 s) + (1000 m)

3. (−120.0 m) = (−5.7 m/s)(6.8 s) + xinitial

a) Write a story or a word problem for which the equation is a solution. There is more

than one possible problem for each situation.

b) Sketch a situation that the mathematical representation might describe.

c) Determine the unknown physical quantity.

4.8 Practice

You are learning to drive. To pass the test you need to be able to convert between different

speedometer readings. The speedometer says 65 mph. (a) Use as many different units as

possible to represent the speed of the car. (b) If the speedometer says 100 km/h, what is the

car’s speed in mph?

4.9 Practice The speed limit on the roads in Russia is 75 km/h. How does this compare to

the speed limit on some US roads of 55 mph?

4.10 Practice Convert the following record speeds so that they are in mph, km/h, and m/s.

(a) Australian dragonfly—36 mph; (b) the diving Peregrine falcon—349 km/h; and (c) the

Lockheed SR-71 jet aircraft—980 m/s (about three times the speed of sound).

4.11 Reason You are moving on a bicycle trying to maintain a constant pace. You cover 23

miles in 2 hours. What is your speed in m/s? If you only rode half of the distance

maintaining the same pace what would the speed be? If you rode 43 miles, what would the

speed be?

4.12 Reason James and Tara argue about speed. James says that the speed is proportional

to the distance and inversely proportional to the time during which the distance was

covered. Tara says that the speed does not depend on the time or distance. Why would each

say what they did? Do you agree with James? Do you agree with Tara? How can you

modify their statements so that you could agree with both of them?

4.13 Hair growth speed Physicists often do what is called “order of magnitude

estimations”. Such estimations are approximate calculations of some quantity that they are

interested in. For example, how do we estimate the rate that your hair grows in mm/s?



29

Think of the following: How often do you get haircuts? How long does your hair grows

during this time? Then convert the time between hair cuts to seconds and the length of your

hair growth to millimeters – then you are almost done. The question is – how will you

report your results? What if after dividing length by time you get a number on your

calculator that looks like 0.005673489? Think how you can report the result so it looks

reasonable.

Did You Know?

Significant digits

When we measure a physical quantity, the instrument we use and the circumstances

under which we measure it determine how precisely we know the value of that quantity.

Imagine that you wear a pedometer (a device that measures the number of steps that you

take) and wish to determine the number of steps on average that you make per minute. You

walk for 26 min (as indicated by your wristwatch) and see that the pedometer shows 2254

steps. You divide 2254 by 26 using your calculator and it says 86.692307692307692. If

you accept this number, it means that you know the number of steps per minute within plus

or minus 0.0000000000000001 steps/minute. If you accept the number 86.69, it means that

you know the number of steps to within 0.01 steps/minute. If you accept the number 90, it

means that you know the number of steps within 10 steps/minute. Which answer should you

use?

The number of the significant digits in the final answer should be the same as the

number of significant digits of the quantity used in the calculation that has the smallest

number significant digits. Thus, in our example, the average number of steps per minute

should be 86, plus or minus 1 steps/minute: 86±1. In summary the precision of the value of

a physical quantity is determined by one of two cases. If the quantity is measured by an

instrument, then its precision depends on the instrument used to measure it. If the quantity is

calculated from other measured quantities, then its precision depends on the least precise

instrument out of all instruments used to measure a quantity used in the calculation.

Another issue with significant digits arises when a quantity is reported with no

decimal points. For example, how many significant digits does 6500 have—two or four?

This is where the scientific notation helps. Scientific notation means writing numbers in

terms of their power of 10. Example: we can write 6500 as 6.5 x 103. This means that the

6500 actually has two significant digits. If we write 6500 as 6.50 x 103 it means 6500 had


three significant digits. Scientific notation provides a compact way of writing large and

small numbers and also allows us to indicate unambiguously the number of significant

digits a quantity has. [This is great too, could add some practice later]

4.14 Evaluate

On the web you might find the flowing statement: “The speed of hair growth is roughly

1.25 centimeters or 0.5 inches per month, being about 15 centimeters or 6 inches per year.

With age the speed of hair growth might slow down to as little as 0.25 cm or 0.1 inch a

month.” Is this result consistent with your estimate? Are the significant figures reported in

for different measurements consistent with each other?

PUM | Kinematics | Lesson 5: How Many Velocities can an Object Have?


31

Lesson 5: How Many Velocities can an Object Have?

5.1 Observe and Reason

Examine the Position vs. Clock Reading graph for the football player below.

a) What were the player’s positions at the points shown with triangles on the graph?

b) Describe the motion of the player in words. Act it out. Pay attention to what

happened at 0 clock reading!

c) James said the football player traveled 5 yards in the negative direction. Tara said

the football player moved 85 yards total. Joe said the football player traveled 5

yards. How did each person arrive at his/her answer?

d) What is the distance traveled by the football player from t = 1.0 s to t = 12.0 s?

What is the path length traveled by the football player travel from t = 1.0 s to t =

12.0 s? What is the displacement of the football player travel from t = 1.0 s to t =

12.0 s?

e) Explain why the values for the quantities in d) are written as t = 1.0 s for example

as opposed to t = 1 s?

32 PUM | Kinematics | Lesson 5: How Many Velocities can an Object Have? © Copyright 2013, June, Rutgers, The State University of New Jersey.

Did You Know?

Position, displacement, distance and path length: Position x is the location of an object

relative to a chosen zero on the coordinate axis.

Displacement x2 - x1 indicates a change in position from clock reading t2 to clock reading t1.

The sign of the displacement indicates the direction of the displacement (+ when the object

moves in the positive direction as x2 > x1 and – when the object moves in the negative

direction of the chosen axis as x2 < x1).

The magnitude of that position change is called the distance. It is always positive.

The path length is the length of the path that the object traveled. If the object returns to the

same point where it started, the displacement and distance are zero, but the path length is

not.

f) What is the distance traveled by the football player travel from t = 2.0 s to t = 19.0 s?

What is the path length traveled by the football player from t6 = 2.0 s to t7 = 19.0 s?

What is the displacement of the football player from t = 2.0 s to t = 19.0 s?

Explain

a) Devise a method for obtaining the value for displacement, distance and path length

on a position versus clock reading graph.

b) Your friend says that to find displacement, he needs to take the position reading at

point 2 and subtract the position reading at point 1. Do you agree or disagree?

5.3 Observe and Reason

Examine the Position vs. Clock Reading graph for the same football player running the

same play below.

a) What were the player’s positions at the points shown with triangles on the graph?



33

b) Examine the graph above. What do you know about this observer compared to the

original observer?


A car stops for a red light. The light turns green and the car moves forward for 3 seconds at

a steadily increasing speed. During this time, it travels 20 meters. The car then travels at a

constant speed for another 3 seconds for a distance of 30 meters. Finally, when approaching

another red light, the car steadily slows to a stop during the next 3 s in 15 meters.

a) What is the total path length that the car traveled?

b) What is the average speed of the car?

Need Some Help?

To find the average speed, you need to divide the total path length traveled by the total

time of travel.

c) How does the total average speed for the entire 9 seconds compare to the average

speed for each of the 3-second intervals? Why are the average speeds different?

Explain.

d) What is the average velocity?

e) A car traveled for 15 s at 10 m/s and another 15 s at 20 m/s. What is the average

speed?

f) The same car traveled 200 m at 10 m/s and another 200 m at 20 m/s. What is the

average speed? Discuss the difference between the results here and in part (e).

Homework


A bus filled with physics students going to Great Adventure for Physics day travels 280 km

West along a straight-line path at an average velocity of 88 km/hr to the west. The bus stops

for 24 min, then it travels 210 km south with an average velocity of 75 km/hr to the south.

a) Diagram and label all the pertinent information for this trip.

b) What is the average velocity for the total trip? What is the average speed for the

total trip?

34 PUM | Kinematics | Lesson 5: How Many Velocities can an Object Have? © Copyright 2013, June, Rutgers, The State University of New Jersey.


The position of an object is represented in the graph above.

a) Describe the motion in words.

b) What is the average velocity of the object for the different time intervals: 0 - 10 sec

and 10 - 20 sec?

c) What is the average speed for the object during the entire 30 sec? What is the

average velocity during that same interval?

d) What is the average speed and average velocity for the time interval from 5 sec – 25

sec?


The picture above is a diagram of a 400m outdoor track. All races begin at the start/finish

line.

Start / Finish

100m

60 m

200 m mark



35

a) If the 1600m race is 4 laps, what is the path length raced? What is the

displacement?

b) The 200 m run begins at the 200 m mark and finishes at the start/finish line, what is

the path length raced? What is the magnitude (the amount) of the displacement, or

distance raced?


In the table on the right you have data about the moving

object. How far did the object travel in the 10 seconds?

I) 230 m; II) 70 m; III) 180 m.

Reflect: What did you learn in this lesson? Why is it important to know

about average velocity and average speed? How is possible for a car to

have an average velocity of zero and average speed of 65 mph? Give an

example.

Time (second) Position (meters)

2 50

4 100

6 100

8 200

10 230

12 300

36 PUM | Kinematics | Lesson 6: Representing the same thing in different ways © Copyright 2013, June, Rutgers, The State University of New Jersey.

Lesson 6: Representing the same thing in different ways

6.1 Observe and Represent

Two cars are let go simultaneously on a smooth floor. You and a friend follow each car and

drop a beanbag every second to mark the location of your car at every time interval.

● ● ● ● ● ● ● ● ● ● ● Car 1

● ● ● ● ● ● Car 2

Describe the motion of each car as fully as possible by answering the following questions:

a) Were the cars ever next to each other? If so, how do you know? At what clock

reading(s) does it happen? Explain.

b) Eugenia answered the question above by drawing the following figure:

● ● ● ● ● ● ● ● ● ● ● Car 1

● ● ● ● ● ● Car 2

Why do you think she circled the dots? How can you help her understand your point of

view?

c) Which direction are the cars moving?

d) Using arrows, how would you make the picture above a more complete

representation of motion?

e) If a general x(t) equation for the first car is x1 (t) = v1 t. What would be the same

about the equation for the second car? What would be different?


Madeline and Gabi walked eastward on a marked path at a velocity of 0.7 m/s. The path

runs East and West with every 1000 m marked. If they begin at the mark of 6000 m and

walked for 25 min, how far have they walked? At what mark would they be at the end of the

walk?


Darshan, roller skating down a marked sidewalk, was

observed to be at the following positions at the times listed

below. Answer the following questions

a) Plot a position vs. time graph for Darshan’s motion.

b) Determine the time it will take for Darshan to skate

to position -85.0m. Be sure to discuss assumptions

made.

t (s) x (m)

0.0 30.0

1.0 26.0

2.0 22.0

5.0 10.0

8.0 -2.0

10.0 -10.0

PUM | Kinematics | Lesson 6: Representing the same thing in different ways


37

c) Draw a dot diagram for the 10-second time interval.

d) How far did Darshan travel in 5 seconds:

I) 10.0 m; II) -12.0 m; III) -20.0 m.


A situation involving constant velocity is described mathematically below.

114mi = (−62mi

hr)(0.35hr) + x

0

a) Sketch a situation that the mathematical representation might describe. There is

more than one possible situation for the equation. Pay special attention to what

happens at the zero clock reading.

b) Write in words a problem for which the equation is a solution.

c) Draw a dot diagram for the motion.

d) Plot a position vs. time graph for this motion.

e) Determine the unknown quantity.


Calculate the average speed and average velocity of a complete round trip in which a train

travels 200 km at 90 km/hr, stops for an 1.0 hour and returns back to the starting point at 50

km/hr.

a) Plot a position vs. time graph for this motion. Identify the place on the graph that

represents the hour break.

b) Solve for as many unknown quantities as possible.

6.6 Practice a) A car moved for 30 min at the speed of 55 mph and for another 30 min at

75 mph. What was the average speed of the car?

b) A car moved for 30 miles at the speed of 55 mph and for another 30 miles at 75 mph.

What was the average speed of the car?

38 PUM | Kinematics | Lesson 6: Representing the same thing in different ways © Copyright 2013, June, Rutgers, The State University of New Jersey.

Homework


A situation involving constant velocity is described mathematically below.

x = (−62.0mi

hr)(0.35hr) + 4.0 mi

a) Sketch a situation that the mathematical representation might describe. There is

more than one possible situation for the equation.

b) Write in words a problem for which the equation is a solution.

c) Draw a dot diagram for the motion.

d) Plot a position vs. time graph for this motion.

e) Determine the unknown quantity.

f) Repeat the same steps for the equation: 114mi = (−vmi

hr)(1.2hr) + (−30.0 mi) .

6.8 Practice (very challenging!)

a) A car moved half of the time at the speed of 55 mph and for the other half at 75 mph.

What was the average speed of the car?

b) A car moved for half the distance at the speed of 55 mph and the other half distance

at 75 mph. What was the average speed of the car?

Here’s an Idea!

Physics is about problem solving. Every problem is different , however some general

strategies might help. Examine the steps below – they might not be relevant for every

problem you encounter, but some of them are useful all the time.

Problem Solving Strategy

Sketch and Translate:

• Read the text of the problem at least three times to make sure you understand what

the problem is saying. Visualize the problem, make sure you see what is happening.

• Sketch the situation described in the problem.

• Include an object of reference, a coordinate system and indicate the origin and the

positive direction.

• Label the sketch with relevant information.

PUM | Kinematics | Lesson 6: Representing the same thing in different ways


39

Draw a physical representation:

• Think of whether a dot diagram or a graph will help you understand the problem.

Represent the problem situation with either or both.

Represent Mathematically:

• Use the sketch(s), diagram(s), and graph(s) to help construct a mathematical

representation (equations) of the process. Be sure to consider the sign of each

quantity.

Solve and Evaluate:

• Solve the equations to find the answer to the question you are investigatin

• Evaluate the results to see if they are reasonable. To do this, check the units, decide

if the calculated quantities have reasonable values (sign, magnitude), and check

limiting cases. Go bck to the sketch and the physical representation to make sure

your answer is consistent with both. Do not rush!

Reflect: What did you learn during this lesson? If you were to write a letter to your

past self about different representations, what would you say?

Why are different representations important in physics? Which representation are

you most comfortable with? Which representation gives you the most trouble? After

you found the latter, make sure you focus on them in the future lessons.

40 PUM | Kinematics | Lesson 7: Using Velocity-versus-Time Graphs to find Displacement © Copyright 2013, June, Rutgers, The State University of New Jersey.

Lesson 7: Using Velocity-versus-Time Graphs to find

Displacement


The figure on the right shows a

velocity-versus-clock reading

graph that represents the motion

of a bicycle, modeled as a point

particle (we are not interested in

the motion of the feet, head,

etc.), moving along a straight

bike path. The positive direction

of the position axis is toward

the east.

a) Describe the motion of

the bike in words.

b) How is this graph

different from the

position versus time

graph for the same motion?

c) Think of how you can use the graph to estimate the bike’s displacement from a

clock reading of 10 s to a clock reading of 15 s. Explain. (Hint: Think of the area of

a rectangle.)

d) Use the graph to estimate the bike’s displacement from a clock reading of 0 s to 20

s.

e) Formulate a general rule for using a velocity versus clock reading graph to

determine an object’s displacement during some time interval if the object is moving

at constant velocity.

f) Is this rule consistent with the mathematical models developed in the previous

lessons x(t) = x0 + vt for constant velocity?


The figure below shows a position-versus-clock reading graph that represents the motion

of a bicycle, modeled as a point like object (we are not interested in the motion of the feet,

head, etc.), moving along a straight bike path. The positive direction of the position axis is

toward the east.

a) Describe the motion of the bike in words.

PUM | Kinematics | Lesson 7: Using Velocity-versus-Time Graphs to find Displacement


41

b) Plot a velocity-versus-clock reading graph for the motion represented in the

position vs. time graph.


1.0

2.0

3.0

4.0

5.0

6.0

20 40 60 80 100 120 140 time (s)

Vel

oci

ty (

m/s

)

Velocity vs. Time


The figure above shows a velocity vs. time graph that represents the motion of a person

moving along a straight hiking path. The positive direction of the coordinate axis is toward

the south.

a) Describe the motion of the hiker in words.

b) Use the graph to determine how far the hiker moved from the clock reading of 10 s

to the clock reading of 25 s. Explain.

c) Use the graph to estimate his distance traveled for the time interval 40 s to 70 s.

d) What is the average speed of the hiker? Explain.

7.4 Evaluate

The following graphs represent the motions of two bicyclists. With which of the statements

about the motions do you agree? Explain your choice.

a) Bike A started moving at constant positive velocity.

b) Bike B climbed over a flat hill.

c) Bike A stopped twice during the trip.

d) Bike B stopped twice during the trip.

e) The last part of the trip bike A was not moving.

f) The last part of the trip bike B was moving at constant speed in the negative

direction.

g) The last part of the trip bike A was moving at constant speed in the negative

direction.

h) When we started observing Bike B it was moving at constant positive velocity.

i) When we started observing bike A it was moving at increasing velocity in the

positive direction, then it reached some constant velocity (positive) and continued

moving for a while, then its velocity started decreasing and it some point it became

zero. The it continued to increased in the negative direction until it reached some

new velocity which it maintained for a while.

x (m) v ( m/s)

t (s) t (s)

PUM | Kinematics | Lesson 7: Using Velocity-versus-Time Graphs to find Displacement


43

j) When we started observing bike B it was moving at constant velocity in the positive

direction, then it stopped for a while, then it started going back to the origin and then

in the negative direction. Finally it stopped.

Here’s an Idea!

Now that you are familiar with different kinds of graphs – position versus time,

displacement versus time and velocity or speed versus time, you might confuse them when

solving problems. To avoid confusion, every time you start working with a graph, slow

down for a moment and say: “Hello Mister Graph. My name is (say your name here). I see

that you are a velocity versus time graph (or a position versus time depending on what

graph it is)!” After you greet a graph like this, you will certainly avoid the confusion. And if

it sounds silly, it’s ok.

Homework


You are driving home from the University of Delaware after a college visit at 65 miles per

hour (mph) for 130 miles. It beings to rain and you slow to 55 mph. You arrive home after

driving for 3 hours and 20 minutes.

a) Diagram and label all the pertinent information for this trip.

b) Plot a position vs. time graph for this motion. (Do not forget to say “Hello Mister

Graph!”)

c) Plot a velocity vs. time graph for the trip. (Do not forget to say “Hello Mister

Graph!”)

What is the average velocity for the total trip? What is the average speed for the total trip?

7.6 Evaluate

Examine the graphs below. Then choose the statements with which you diagree. Explain

your reasons. Then explain why someone would choose those wrong answers as correct.

A B C D

a) The first two graphs (A and B) provide the same information.

b) The second two graphs (C and D) provide the same information.

v, m/s

t, s 1 2 3 4

30

20

10

-10

-20

v, m/s

t, s 1 2 3 4

30

20

10

-10

-20

x, m

t, s 1 2 3 4

30

20

10

-10

-20

x, m

t, s 1 2 3 4

30

20

10

-10

-20


c) Object A traveled 60 meters in 3 seconds from the location it was at the 0 clock

reading.

d) Object B traveled 40 meters in 2 seconds in the positive direction.

e) Object C was not moving during the experiment.

f) Objects A and D were not moving during the experiment.

g) Object C was moving in the negative direction at the speed of 20 m/s.

h) Object D was moving in the negative direction at the speed of 20 m/s.

i) Object C was moving in the negative direction at the speed of (-20 m/s).

j) Object D traveled 40 m in 2 seconds in the negative direction.

7.7 Pose your own problem Make up a problem to solve which one needs to know how to

calculate displacement of an object from a velocity versus time graph. Solve the problem

and make a list of difficulties that your friends might have with it. (Hint: to start, you can

modify one of the problems in the lesson.)

Reflect: What did you learn in this lesson? What was easy? What was difficult? What

could you have done differently “to ease the pain”?

PUM | Kinematics | Lesson 8: Lab: Will the Cars Ever Meet?


45

Lesson 8: Lab: Will the Cars Ever Meet?

8.1 Design an Experiment

The goals of this experiment are:

• to apply kinematics equations to describe real-life phenomena;

• to become aware of assumptions that one makes when applying mathematical

equations to a real situation;

• to understand that one cannot avoid experimental uncertainties with particular

equipment.

In this experiment, you will:

• determine the kind of motion two cars have, and

• use kinematics equations to predict the location of where the two cars will meet if

they start at given locations and move toward each other.

Equipment: Two battery-operated cars, meter stick, stopwatch, sugar packets or any kind of

markers for the cars locations

Setting up the experiment: First, experiment individually with the cars to determine whether

each car moves with constant speed or changing speed and then determine the speed of each

car. You can use the methods you learned in the first lab to record positions of the cars

every second. Use the following steps:

a) Draw a clearly labeled diagram of your experimental setup.

b) List the physical quantities you will measure and how you will measure them.

c) Perform the experiment. Record the data in appropriate formats, such as a dot

diagram, a table, and a graph.

d) What is the uncertainty in your measurements for clock reading and position?

Which uncertainty is the largest in your experiments?

e) How can you represent the uncertainty on your graph? Does it change the way that

you would draw the trend lines?

f) Find the speed of each car.

Did You Know?

Experimental uncertainty

When we collect data there are two kinds of uncertainty. First, instrumental uncertainty is

due to the fact that the scales of the instruments have divisions. We cannot measure

anything more precisely than half of the smallest division. We will learn more about this

46 PUM | Kinematics | Lesson 8: Lab: Will the Cars Ever Meet? © Copyright 2013, June, Rutgers, The State University of New Jersey.

uncertainty later in the lesson as in this particular experiment another type of uncertainty is

more important. It is called random uncertainty which is due to the fact that sometimes it is

difficult to collect data consistently. For example, when you measure how much time it

takes for one of your cars to move 2.0 m and obtain a particular value for the time interval,

there is little guarantee that next time you repeat the same experiment you will get exactly

the same value. It is due to the fact that you cannot release the car the same way in both

experiments, or that the car encounters a bump on the floor, or some other reason. To find

the random uncertainty, you need to measure the same value several times (at least three) –

let’s say you measured n1; n2; n3. Next you find the average of three numbers

n =n1 + n2 + n3

3. Then you find how far in terms of the magnitude each measurement is

from the average: ∆n1 = n − n1 ; ∆n2 = n − n2 ; ∆n3 = n − n3 and finally choose the largest

of the three. This is the largest uncertainty in the value you are trying to determine. Now

you can say that the value of your measurement is where ∆nlargest is the

largest random uncertainty of your measurement. Sometimes it is more useful to express the

uncertainty not as an absolute number but as a percent uncertainty (

∆nlargest

n�100%).


This time you need to predict where the cars would meet if they were placed 3 m apart

initially and started at the same time. The steps outlined below will help you with reasoning

through this prediction.

a) Think of how you can represent the motion of each car with the equation x(t).

Remember that the cars start at two different locations and move in different

directions!

b) Use the equations that you wrote to predict where the cars will meet if released

simultaneously. How certain are you in the location?

Did you know?

Instrumental Uncertainty

To express your prediction as a range and not as a single number: x = x1 ± the value of the

uncertainty, you need to think of how you know what the uncertainty is. You have not

performed the experiments, so there random uncertainty is not known. However, we as we

talked before, every instrument has the uncertainty due to the size of its smallest division.

For the ruler it might be 0.5 mm. If you are using more than one instrument – for example a

watch and a ruler, then the instrument that brings in the biggest uncertainty will determine

your final instrumental uncertainty. For example – the watch has an uncertainty of 0.5

seconds and a ruler has an uncertainty of 0.5 mm. How can you compare mm to seconds?

To answer this question we will use the weak link rule. This is how it works. We estimate

the relative uncertainty that each instrument gives us – this is the half of the smallest



47

division divided by the smallest measurement you make with this instrument and compare

to the relative uncertainty due to the second instrument. The instrument with the largest

relative uncertainty will determine the instrumental uncertainty of your experiment. For

example you measure 3 m length with the ruler that has 1 mm divisions, thus your relative

uncertainty for the ruler is 0.5 mm/3 m = (0.5 x 10-3

m)/ (3m) = 0.2 x 10-3

(or 0.02% of the

measurement); you also measure 10 seconds with your watch, the uncertainty is 0.5 s / 10 s

= 0.05 (or 5 % of the measurement). This is much larger than the 0.2 x 10-3

relative

uncertainty by the ruler. Thus if you use both instruments, you need to report your results

(or predictions) within the interval of 0.05 times (5%) the measured value: for example:

instead of 3 meters, we will need to write 3 m ± (0.05 x 3 m) = 3 m ± 0.15 m.

Actually, you can use the weak link rule to evaluate any uncertainties in your result: first

you evaluate random uncertainty as the percent of the result, then the instrumental –

whichever is larger will give you the estimate of the uncertainty your result has.

c) When you make the prediction, think of the assumptions that you are using. One of

them is that the cars move at constant speed right from the start of the measurement.

How will this assumption affect the meeting location? What are other assumptions

you are making?

d) Show your prediction to your instructor and then try the experiment. Record the

outcome. Make sure that your experiment allows you to evaluate random uncertainty

and compare it to the instrumental.

e) Reconcile any differences in your prediction and the outcome of the experiment. Be

specific. Can you explain any differences because of experimental uncertainties or

assumptions that might not be valid?

Homework

8.3 Write a lab report

Write a scientific report about the two experiments that you performed. The report should

describe everything you did so that a person who did not do the experiments could repeat

them and get the same results. Make sure that you justify your judgment about the outcome

of the experiment.

Use the rubrics to improve your lab report. Make sure you study each of the abilities and

decide whether you can improve your report based on the rubrics.


Ability to collect and analyze experimental data

Scientific Ability Missing An attempt Needs some

improvement Acceptable

1

Is able to

identify sources

of experimental

uncertainty.

No attempt is

made to

identify

experimental

uncertainties.

An attempt is made

to identify

experimental

uncertainties but

most are missing,

described vaguely,

or incorrect.

Most experimental

uncertainties are

correctly identified

but the source of

the biggest

uncertainty is not

specified.

All experimental

uncertainties are

correctly identified

and the source of

the biggest

uncertainty is

specified.

2

Is able to

specifically

evaluate how

identified

experimental

uncertainties

may affect the

data.

No attempt is

made to

evaluate

experimental

uncertainties.

An attempt is made

to evaluate

experimental

uncertainties but

most are missing,

described vaguely,

or incorrect. Or the

final result does not

take the uncertainty

into account.

The final result

does take the

identified

uncertainties into

account but is not

correctly evaluated.

The experimental

uncertainty of the

final result is

correctly evaluated;

the final result is

written within the

margin of

uncertainty.

3

Is able to record

and represent

data in a

meaningful way.

Data are

either absent

or

incomprehens

ible.

Some important data

are absent or

incomprehensible.

All important data

are present but

recorded in a way

that requires some

effort to

comprehend.

All important data

are present,

organized, and

recorded clearly.

4

Is able to

identify the

assumptions

made in using

the

mathematical

procedure.

No attempt is

made to

identify any

assumptions.

An attempt is made

to identify

assumptions, but the

assumptions are

irrelevant or

incorrect for the

situation.

Relevant

assumptions are

identified but are

not significant for

solving the

problem.

All relevant

assumptions are

correctly identified.

5

Is able to

specifically

determine the

ways in which

assumptions

might affect the

results.

No attempt is

made to

determine the

effects of

assumptions.

The effects of

assumptions are

mentioned but are

described vaguely.

The effects of

assumptions are

determined, but no

attempt is made to

validate them.

The effects of the

assumptions are

determined and the

assumptions are

validated.


a) Several motions are described mathematically below. All quantities are in SI units

(position in meters and time in seconds). Represent these motions in words and

graphically as x(t) and v(t). Draw the position and velocity versus time graphs

without creating a data table; use the information about the slope and the y-intercept

to draw the graphs. Make sure that you draw all the x(t) graphs using the same set of

axes and all the v(t) graphs using the same set of axes (make sure you say Hello

Mister Graph to each of the graphs when you start drawing them)..

x = +3t + 5 x = (-3t) + 5 x = (-3t) +(– 5)

x = 3t +(– 5) x = 3t x = -3t



49

b) If you choose a reference frame in which a toy car is traveling at a constant speed

and then you decide to change the reference frame to a different one, will the car

still be traveling at constant speed? Explain.

c) The motion of two objects are described by two x(t) functions: x = (-12) + 3 t and

x* = +24 +(– 7 t). All quantities are in SI units. Describe the motion of the two

objects in words and with graphs of x(t). Use the equations to determine where and

when the objects will meet. What assumptions did you make?


Consider the graphs below for objects A and B.

How does the motion of the object A in graph 1 compare to that of A in graph 2?

a) Say “Hello Mister Graph to the graphs.” What do graphs represent?

b) How does the motion of object B in graph 1 compare to the motion of object B in

graph 2?

c) Which object has the smaller speed in graph 2? Explain.

d) Describe what is happening at the point where the function of object A intersects

that of object B.

e) Which object traveled a greater distance during the first 6 seconds in graph 1?

Explain.

f) Write a function x(t) for all four objects.

Time (s) 6.0 sec

Po

siti

on

(m

)

Time (s) 6.0 s

Po

siti

on

(m

)

A

B

B

A

Graph 1 Graph 2



A New Jersey Transit train leaves Bay Head and heads north to Secaucus. Another train

leaves Secaucus and heads South to Bay head, both trains are express trains (which means

they do not stop). If the northbound train has an average velocity of 35 mi/hr and the

southbound has an average velocity of 52 mi/hr. The distance between the two stations is 60

miles, predict where the two trains will meet. Be sure you address the assumptions you

made in order to solve this problem.


A student starts walking at 5 ft/s in a corridor A and is 20 ft away from the intersection of

corridors A and B. A second student starts at the same time running at 8 ft/s in corridor B

and is 32 feet away from the intersection.

a) Create a dot diagram for the problem.

b) Graph the motion of the two students on a position-versus-time graph

c) Represent the motion of each student with a function.

d) How many different functions can you write for motion of each student? What will

be different and what will be the same?

e) Will the two students collide? Show your work and explain your reasoning.

8.8 Reason Examine the graph.

What is the object’s average speed during the first

4 seconds of its trip?

I) 17.5 m/s; II) 4.4 m/s (17.5/4); III) 8.8 m/s (17.5/2)

x, m/s

t, s 0 1 2 3 4 5 6

20

15

10

5

0

PUM | Kinematics | Lesson 9: Motion Diagrams


51

Day 1 Day 2 Day 3 Day 4 Day 5

Lesson 9: Motion Diagrams

9.1 Describe

During art class you made a clay figure of a person and the art teacher was very impressed

by your work. She even displayed your work for the whole class. However throughout the

week you noticed that people were changing your artwork!

For each day, determine if you artwork was stretched up, squished down, or unchanged.

Then draw an arrow to represent the direction of change. Make sure your arrow shows the

amount of stretching or squishing.

Original to Day 1: Day 1 to Day 2:

Day 2 to Day 3: Day 3 to Day 4:

Day 4 to Day 5:

Here’s An Idea!

This activity may not have seemed like physics but it is design to help you understand

change arrows. This activity serves as an analogy for you to refer back to when you’re

having trouble. Scientist often will make analogies for complex systems in order to better

understand what may be occurring.

52 PUM | Kinematics | Lesson 9: Motion Diagrams © Copyright 2013, June, Rutgers, The State University of New Jersey.


Use a small ball and a long tilted ramp at a very small angle. Let the ball roll down the

ramp.

a) Draw a dot diagram that shows the velocity during each time interval, direction of

motion, and a directional axis.

Need Some Help?

What you need to draw is something physicists call, a motion diagram. It is a sophisticated

replacement for a dot diagram that conveys more information about a situation. If you are

new to this representation, you may want to list or label the important features so that you

are sure to include these when you draw one for yourself.

Example:

b) What does the length of the arrow tell you about the motion of the object?

c) What two ways is direction indicated on the representation? Why are both

necessary?

d) What does the length of each arrow tell you about the motion of the car at a

particular dot or position?

e) What does the length of each arrow tell you about the motion of the ball at a

particular time?

f) With a classmate, act out the motion represented in the motion diagram.

g) If you were to make a position versus time graph for this motion how would the

trend line look? Why does this make sense?

9.3 Observe and Represent

v2 v3 v4v1

v2 v3 v4v1



53

In this activity, we’re going to examine the motion diagram provided in the previous

activity.

a) Think about the previous activity. What must you do to velocity arrow one to get

velocity arrow two? What direction? How much?

b) Answer the questions in part (a) for arrows 2 and 3 as well as arrows 3 and 4. Is

there any difference?

Did you know?

The arrows that you drew to show to difference between velocity arrows are called ∆∆∆∆v

arrows or change in velocity arrows. We can line them up and compare the size of the

arrows in order to determine the change. These arrows tell us the direction and the

magnitude (size) of the change. It’s just like the direction and amount of

squishing/stretching from the previous activity! A complete motion diagram includes ∆v

arrows.

Example:

c) What does the ∆∆∆∆v arrow tell you about the motion of the object? Explain.

d) Revisit the dot picture you drew for the ball rolling down an incline. Create a

complete motion diagram for the motion of the ball.

9.4 Reason and Explain

Describe the motion represented in this motion diagram.

a) Is the object speeding up, slowing down, or moving at constant velocity? How do

you know?

b) What direction should the ∆v arrow be pointing? Explain how you determined this.

c) Is the change in motion (∆v arrows) in the same direction that the object is moving

(v arrows)?

d) Make a rule for speeding up and slowing down by comparing motion (velocity

arrows) to change in motion (∆v arrows)

v2v1

v2

v1 The second v arrow is longer

than the first so it’s a stretched

to the right (positive direction)

and shows speeding up.

v2v3v4 v1

54 PUM | Kinematics | Lesson 9: Motion Diagrams © Copyright 2013, June, Rutgers, The State University of New Jersey.

Did You Know?

Constructing a Motion Diagram: Here is a tool to help you learn how to construct motion

diagrams. Motion diagrams provide a concrete way to represent motion.

Motion diagrams help you to represent, visualize, and analyze motion. They are especially

useful for checking the quantitative math descriptions of motion that you will learn later.

Homework


a) Make up a story about the motion of some object and represent it with a motion

diagram.

b) Draw motion diagrams for: a car starting to speed up from rest next to at a street

light, for the car coasting along a street with a constant speed passing the street light

along the way, and for the car slowing down to a stop next to a street light.

c) Draw a picture of the three situations above and then sketch a position-versus-time

graph for each. Assume that the streetlight is the origin.

d) Draw a picture of the three situations above and then sketch a position-versus-time

graph for each. Assume that the streetlight is the origin.


The table below describes experiments for the motion of a ball. For each experiment,

something happens to the ball and its motion changes. Visualize every experiment and if

you have a ball – perform it! Use the information in table and your

experiments/imagination to fill in the blanks in the table below.

Draw dots to

represent the

position of the

object for equal

time intervals.

Point arrows in the direction of

motion and use the relative lengths

to indicate how fast the object is

moving between the points.

Draw arrow to indicate how

the arrows are changing.

Draw the arrows thicker

than the arrows.

v1 v2 v3



55

Initial motion

∆�v

Final motion

Motion diagram to

match

a) Not moving –x direction

b) Moving in –y

direction

Moving faster in

–y direction

c) Moving in –x

direction

Not moving in x

direction

d) –x direction Not moving in x

direction

e) Moving in +x

direction

+x direction

f) +y direction Moving in –y

direction


The table below describes the velocity of a ball for different experiments. For each

experiment, something happens to the ball and its velocity changes. Use the information in

table to fill in the blanks in the table below.

Initial velocity,

�v

i

∆�v Final velocity,

�v f

Motion diagram to

match

a)

b)

c)

d)

e)

Reflect: What did you learn in this lesson? How is motion diagram similar

to a dot diagram? How is it different?

56 PUM | Kinematics | Lesson 10A: Free Falling? © Copyright 2013, June, Rutgers, The State University of New Jersey.

Lesson 10A: Free Falling?

10.1 Test your Idea

In front of the class, drop an object such as a tennis ball. Observe the motion of the ball as it

falls.

a) As a class, discuss the motion of the falling object. What do the students think about

the motion and why?

b) Design an experiment that will allow you to use the derived kinematics equations to

predict the time interval if you dropped the object from the bleachers (or some other

high place). In designing your experiment what assumptions are you making about

the motion?

c) Before experimenting, discuss the outcomes. If the outcome is the same is as the

predicted time, what does this mean about the kinematics expression and the

assumption(s)? If the outcome is different than the predicted time, what does this

mean about the kinematics expression and the assumption(s)?

Rubric to self-assess your prediction

Scientific

ability

Missing An attempt Needs some

improvement

Acceptable

Is able to

distinguish

between a

hypothesis

and a

prediction

No prediction is

made. The

experiment is not

treated as a testing

experiment.

A prediction is

made, but it is

identical to the

hypothesis.


and is distinct from the

hypothesis but does not

describe the outcome of

the designed

experiment.

A prediction is made,

is distinct from the

hypothesis, and

describes the outcome

of the designed

experiment.

Homework

10.2 Reflect

Reflect on the last nine lessons while considering the mathematical model x(t) = x0 + v(t).

Summarize the assumptions made while using this model.


Go to http://paer.rutgers.edu/pt3/experiment.php?topicid=2&exptid=38 on the video

website and collect position versus time data for the falling object. Use the data to

draw a dot diagram for the falling ball. What can you say about its motion based on the

diagram?

PUM | Kinematics | Lesson 10B*: Review of Graphical Representations


57

Lesson 10B*: Review of Graphical Representations

10B*.1 Hypothesize

Let’s review position versus time graphs. Use the ideas developed from the previous lessons

to help you develop the following rules.

a) What does constant pace motion look like on a position versus time graph?

b) What does speeding up motion look like on a position versus time graph?

c) What does slowing down motion look like on a position versus time graph?

10B*.2 Test Your Idea

Use your newly modified hypotheses from the previous activity to predict how you’d have

to move so that a motion detector creates position versus time graphs that match the

previous graphs. Explain how your prediction compares to the outcome.

10B*.3 Reason

Examine the graphs below and then answer each of the questions below by recording the

associated letters on the line provided. The units for time are seconds.

a) Which graphs represent objects moving at constant pace? ________________________

b) Which graphs represents objects speeding up? _________________________________

c) Which graphs represent objects slowing down? ________________________________

d) Which graphs represent an object moving in the negative direction? ________________

e) Do any of the graph show an object that is not in motion? How do you know? Can we

consider this a constant pace?

58 PUM | Kinematics | Lesson 10B*: Review of Graphical Representations © Copyright 2013, June, Rutgers, The State University of New Jersey.

a b c

d e f

PUM | Kinematics | Lesson 11: Part I Speeding index


59

Lesson 11: Part I Speeding index

I. Speeding Up Index

Today you will look at another kind of index. Your task is to come up with a

speeding up index for cars. You will see pictures of several cars on the next page, and

you need to come up with one number to stand for each car’s speeding up.

There is no watch or clock to tell you how long each car has been going.

However, all the cars drip oil once a second. (They are not very good cars!) The

speedometer reading tells you how fast the car is going when the oil drips.

This task is a little harder than before.

• A company always makes its cars speed up in the same way.

• We will not tell you how many companies there are.

• You have to decide which cars are from the same company. They may look

different!

To review:

(1) Make a speeding up index for each car. A bigger index means a car speeds up

more.

(2) Decide how many companies there are.

(3) To show the cars that are from the same company, draw a line that connects the

cars.

Reasoning Questions:

1. Use everyday language to describe the specific information that the speeding up

index tells you about the car’s motion.

2. How many car companies are there? How do you know?

3. It can be easy to confuse fast with speeding up quickly, or slow with speeding up

slowly. What is the difference? Use the following questions to help sort this out:

a) Which car is the fastest? Explain what “fastest” means.

b) How long will it take the fastest car to speed up by 75 mph?

c) Which car is the slowest? Explain what “slowest” means.

d) How fast will the slowest car be going 10s after it starts?

Oil drops

5 mph

60 PUM | Kinematics | Lesson 11: Part I Speeding index © Copyright 2013, June, Rutgers, The State University of New Jersey.

e) Compare “fast” to “speeding up quickly” and “slow” to “speeding up slowly.”

Can a car be going fast and speed up slowly? Can a car be going slow and speed

up quickly? Explain.

5 m

ph

5

0

57

.

45

mp

h2

5 m

ph

5 m

ph

55

mp

h1

0 m

ph

34

mp

h2

4 m

ph

14

mp

h4

mp

h

72

mp

h1

2 m

ph

9 m

ph

15

mp

h0

mp

h

Car

E

Car

D

Car

C

Car

B

Car

A

5 m

ph

5

0m

ph

57

mp

h

PUM | Kinematics | Lesson 11: Part I Speeding index


61

6 m

ph

35

mp

h2

5 m

ph

15

mph

5 m

ph

50

mph

35

mp

h20

mp

h5

mp

h

34 m

ph

24

mp

h1

4 m

ph

4 m

ph

57

mp

h42

mp

h2

7 m

ph

12

mp

h

9 m

ph

3 m

ph

0 m

ph

Car E

Car D

Car C

Car B

Car A

62 PUM | Kinematics | Lesson 11 Part II: Motion of a Falling Object © Copyright 2013, June, Rutgers, The State University of New Jersey.

Lesson 11 Part II: Motion of a Falling Object

11.1 Observe and find a pattern

Vernier Software – Video

http://paer.rutgers.edu/pt3/experiment.php?topicid=2&exptid=38 or tinyurl.com/3akwvjy

Use the video of the falling ball to collect position versus time data. Notice that for a few

frames the data are missing, it should not affect your results.

a) Describe the motion of the ball.

b) Decide on an origin and measure the position of the ball at the beginning of each

time interval. How did you decide how to measure the position?

c) What is the instrumental uncertainty in your measurements?

In case you cannot access the videos, a few data points are provided in the table below

Clock Reading (s) Position (m)

0.00 0

0.10 0.02

0.20 0.19

0.30 0.44

0.40 0.8

0.50 1.27

Need Some Help?

More about the instrumental uncertainty For example, you are measuring the length of

the table to be 2m with a ruler that has centimeter divisions. The half of the smallest

division is 0.5 cm. You can write your measurement as l = 2.000 m ± 0.005m . 0.5 cm

constitutes 0.25% of 2 m, thus we can also write the measurement as l = 2.000 m ± 0.25% .

The latter way of writing the instrumental uncertainty explains why using the same

instrument to make larger measurements gives you a more precise result than making

smaller measurements (for example 0.5 cm is 0.25% of 2 m but it is 10% of 5 cm, thus

when you measure the distance of 5 cm with this ruler, you have 10% uncertainty).

d) Record the position and time data for the ball in a table. Can you see any patterns?

Explain.

PUM | Kinematics | Lesson 11 Part II: Motion of a Falling Object


63

e) Plot a position versus time graph for this object. What type of function does the

trend line resemble? Does this graph represent an object traveling with constant

velocity? How do you know?


a) Now we will examine graphically how the velocity changes. Calculate the average

velocity for the ball for each time interval and complete the table that follows. What

is the uncertainty for the velocities? (You may use the table of data in activity 11.1

or collect your own data)

Time interval

∆ t =tn – tn-1

Displacement

∆ x = xn – xn-1

Average clock

reading

(tn + tn-1)/2

Average velocity

b) What patterns do you notice in the table, what do these patterns indicate about the

motion of the falling ball?

c) Plot an average velocity versus time graph. Write a function for how the speed

changes with time, v(t).

d) What is the meaning of the slope of this line? Think about the physical meaning of

the slope of the line and name it.

e) Discuss how the equation would change if the ball were slowing down instead of

speeding up. Discuss how your equation would change if you the ball were initially

thrown upwards.

f) As we have seen, the notion of a rate of change is an important mathematical and

scientific idea. In this case, the rate at which the velocity of an object changes is

referred to as the acceleration of the object. The rate at which the distance traveled

by an object changes is called the velocity of the object.

Did you know?

Acceleration of an object moving at constant acceleration is the slope of the velocity versus

time graph and is equal to the change in velocity of the object divided by the time interval

during which this change in velocity occurred. When the object is moving at constant

acceleration this ratio is the same for any time interval


a =v2 − v1

t2 − t1

where v2 – v1 ( ∆v ) is any change in position during the corresponding time interval t2 – t1

( ∆t ). The unit for acceleration is m/s/s or m/s2.

11.3 Practice

Imagine a car moving at a speed of 10 m/s. It starts speeding up at an acceleration of 2

m/s/s. Do not use any formulas, just your understanding of acceleration, to decide what the

car’s speed will be after 2 second; 4 seconds, 6 seconds.

11.4 Practice

Imagine a car moving at a speed of 10 m/s. It starts slowing down at an acceleration of 2

m/s/s. Do not use any formulas, just your understanding of acceleration, to decide what the

car’s speed will be after 2 second; 4 seconds, 6 seconds.

11.5. Reason

a) You know that velocity has magnitude (called speed) and direction. What about

acceleration? If you think it does, how can you argue your point? If you think it does not,

how do you argue your point? Find a student in class who disagrees with you and try to

discus this issue with her/him.

b) How is acceleration related to the velocity change arrow on the motion diagram?

Explain.


The motion diagrams in the illustrations below represent the motion of different objects

modeled as point-like objects. The arrows are velocity arrows.

A different coordinate axis is provided in each situation. An open circle indicates the

location of interest.

1. Draw velocity change arrows on each diagram above.

x 0 I

x 0

I

II

y

0

II

I II

I



65

2. Fill in the table that follows. Be sure to make your choices relative to the coordinate

axis shown with each motion diagram.

Describe the motion in words. Determine the

sign (+, 0, or –)

of the position.

Determine the

sign (+, 0, or –)

of the velocity.

Determine the sign

(+, 0, or –) of the

acceleration.

a)

Location I:

Location I: Location I:

b)

Location I:

Location II:

Location I:

Location II:

Location I:

Location II:

c)

Location I:

Location II:

Location III:

Location I:

Location II:

Location III:

Location I:

Location II:

Location III:

11.7 Reason

What does it mean that the velocity of a bicyclist is -10.0 m/s? What does it mean if this

bicyclist is accelerating at 2 m/s/s? at -2 m/s/s?

11.8 Evaluate

Tara says that if an object has negative acceleration, it is slowing down. Do you agree or

disagree? How can you argue your point of view?


The graph below represents the motion of a person riding a moped over a period of time as

seen by the observer on the ground. South is denoted by positive numbers.

a) What kind of information does this graph represent? (Do not forget to say “Hello

Mister Graph!”)

b) What was the speed of the moped at the beginning of observations? What was the

velocity? What was the speed at 25 s clock reading? What was the velocity?

c) What happened to the speed between 20 and 44 seconds? Between 44 and 65

seconds? When was the moped at rest? When did it climb a hill?


d) What accelerations did the moped have during the trip? Provide numbers (include

the signs).

e) Compare the distance traveled during the first 20 seconds and the second 20

seconds. Which one is bigger? How do you know?

f) For the clock reading of 80 seconds choose the answer that you think is correct:

I) The graphs is flat, so the moped must have stopped.; II) The moped’s speed was

0.15 m/s (12 divided by 80); III) The moped’s seed was 0.34 m/s (12 divided by 35);

IV) the speed was 12 m/s (the graph shows velocity vs time).

Homework


Study the following three graphs taken from actual laboratory data. Determine the

acceleration for the motion represented on each graph. What does it mean if the acceleration

is positive? Negative?

Write the functions v(t) for the following graphs (on the next page).



67

a) Describe a

situation

that will

match each

graph.

b) Compare

the velocity

versus time

graphs for

objects that

move at

constant

velocity

and objects

that move

with

changing

velocity.

c) What can

you say

about the

acceleratio

n of the

moving

objects in

the graphs?

Explain.

In the following

problems use the

problem solving

strategy!

11.11 Practice

On a bumper car ride, friends smash their cars into each other (head-on) and each has a

speed change of 3.2 m/s. If the magnitudes of accelerations of each car during the collision

averaged 28 m/s2, determine the time interval needed to stop for each car while colliding.

Specify your reference frame.

11.12 Practice


A bus leaves an intersection accelerating at +2.0 m/s2. What is the speed of the bus after 5.0

s? What assumption did you make? Explain.

11.13 Practice

What was the speed of a jogger if she needed 2 seconds to stop at the acceleration of 3

m/s/s. Is the answer realistic?

11.14 Estimate

Estimate the acceleration of a car one of your family members drives. How did you make

the estimation? How many significant digits do you have in your result?

11.15 Practice

A jogger is running at +4.0 m/s when a bus passes her. The bus is accelerating from + 16.0

m/s to + 20.0 m/s in 8.0 s. The jogger speeds up at the same acceleration. What can you

determine about the jogger’s motion using these data?


A person on a motorcycle facing South is initially at rest. The motor cycle speeds up from

rest to 15 m/s in three seconds. For the next 8 seconds, the motorcycle travels at the same

speed. Over the next 12 seconds the motorcycle slows to rest uniformly. Plot a

acceleration-versus clock reading graph for the motor cycle. What assumptions did you

make about the motorcycle and its motion?


The figure below shows a velocity-versus-clock reading graph that represents the motion

of an object, modeled as a point particle (we are not interested in the motion of the feet,

head, etc.), moving along a straight path. The positive direction of the position axis is

toward the North.



69

a) Describe the motion of the object in words.

b) When does the object travel at constant velocity? Explain how you know.

c) When does the object have a velocity equal to zero? Explain how you know.

d) What is the average acceleration for the first three seconds?

e) What is the average acceleration for the time interval 6.0 – 9.0 seconds?

f) Plot an acceleration-versus-clock reading graph for the motion of the object.

g) Examine 4.5 seconds, what can you say about the object’s acceleration and velocity

at that clock reading?


The false-color image on the next page shows the Gangotri Glacier, situated in the

Uttarkashi District of Garhwal in the Himalayas. Currently 30.2 km long and between 0.5

and 2.5 km wide, Gangotri Glacier is one of the largest in the Himalayas. Gangotri has been

receding since 1780.

Use the satellite image to estimate the speed at which the glacier is receding. Describe your

procedure and collect all data from the image.


Reflect: What did you learn during this lesson? How acceleration is

similar to velocity? How is it different? Why do you need both?

PUM | Kinematics | Lesson 12: Position of an Accelerating Object as a Function of Time


71

Lesson 12: Position of an Accelerating Object as a Function of

Time

12.1 Hypothesize (Derive a Mathematical Model)

Recall the initial position and clock reading data from the previous lab. When considering

the motion of the falling ball, how is position related (mathematically) to time?

Use the method developed in lesson 11 to find a relationship between the displacement of

an object (moving with constant acceleration during some time interval), its velocity at the

beginning of this interval, its acceleration, and the length of the time interval. Start by

drawing a velocity-versus-clock reading graph and examining the area under the graph line.

Express the relationship in terms of v0, a, and t.

Use the expression for the displacement to write the function x(t) for the object moving at

constant acceleration. Write the function in terms of x0, v0, a, and t.

12.2 Test Your Idea with Phet Simulations

Go to http://phet.colorado.edu/en/simulation/moving-man

and click on Run Now!

Click on the Charts tab in the upper left corner of the

window. You should see a screen like the one shown.

There is a man at top of the simulation who can move 10 m

in either direction from the origin. The simulation also

includes axes of position, velocity and acceleration graphs

that will reflect his motion. Since you are not going to use

the acceleration graph right away, you can close it by

clicking on the small window in the upper right hand corner of each section.

To eliminate the walls, click on “special features,” then click on “free range”.

Use the hypothesized mathematical model in activity 12.1 to predict the position.

Scenario 1: The man’s initial position is at the tree where he is initially at rest. He has an

acceleration of 0.75 m/s/s to the right.


b) Create a position vs. time and velocity vs. time graph for this function.



consistent?

d) Predict the location of the moving man after 5 seconds. Show your work

72 PUM | Kinematics | Lesson 12: Position of an Accelerating Object as a Function of Time © Copyright 2013, June, Rutgers, The State University of New Jersey.

e) Perform the experiment by entering given quantities in the respective simulation

boxes and click Go! Compare your predicted value to the outcome of the testing

experiment. Do they agree or disagree? If they disagree, revise your mathematical

model of the moving man’s motion.

Scenario 2: The man is walking initially at 0.75 m/s towards his home starting from the

position of 7m to the left of the origin. At this point, he begins to increase his velocity at a

rate of 0.2 m/s every second.





consistent?

d) Predict the time when he arrives at the origin.




Scenario 3: The man starts at the 5m mark by the house and is walking towards the 1.0 m/s

towards the tree. He is accelerating towards the tree at 0.5 m/s/s





consistent?

d) Predict the position when the man is moving at a speed of 5 m/s.




Scenario 4: The man starts at the house running at 7.0 m/s towards the tree. He is slowing

up at 1.0m/s/s.





consistent?



73

d) Predict the position of the man’s final position when he comes to rest.




12.3 Practice

Velocity of a motorcycle changes according to the graph below.

a) What do the slopes of the line segments on the graph tell you? When does the motorcycle

have a positive acceleration, zero acceleration, negative acceleration?

b) What is the distance traveled during the first 20 seconds? From 20 to 45 seconds? From

45 to 65 seconds? From 65 to 85 seconds? From 85 seconds to the end of the recorded trip?

c) Determine the total path length traveled and the displacement of the motorcycle.

Make sure you use the Problem Solving Strategy for the problems below.

12.4 Practice

A bus leaves an intersection accelerating at +2.0 m/s2 from rest (think what the term rest

means). Where is the bus after 5.0 s?

12.5 Practice

A bicyclist slowed down from 8 m/s to 2 m/s in 3 seconds. What was the acceleration of

the bicycle? How far did it move during this process?

12.6 Reason

The motion of a car can be described by the following function. All quantities are in SI units

x(t) =14t + 3t2


Explain the meaning of each number. Describe the motion in words, with a motion diagram,

and with a picture with a reference frame.

a) What would the v(t) expression look like?

b) Determine the position and the velocity of the car after 5 seconds.

Another car’s motion is describe by the following equation:

x(t) = (-12) + 7t + (– 0.4t2)

c) How does this motion compare with the previous car? Repeat (a) and (b) for this

object.

d) Act out each motion. To act out, have two students to represent each moving

objects. Let classmates give those students directions on how to move.

12.7 Compare and Contrast

Jim says: “We learned so many different words: constant velocity, zero velocity, constant

acceleration and zero acceleration. I do not understand the difference between them, all

sound like motion to me”. Do you feel similar to Jim? If you do, it is normal. To help

yourself navigate through the new ideas, work through the following exercises.

Draw a motion diagram for each scenario to help you construct each situation.

a) Describe a situation when an object moves with an acceleration equal to zero and a

velocity that is a non-zero negative number.

b) Describe a situation when an object moves with an constant positive acceleration

and velocity is positive number.

c) Describe a situation when an object travels with a constant negative acceleration and

an positive velocity.

d) Describe a situation when an object travels with a constant negative acceleration and

a negative velocity.

12.8 Reason

The driver of a car moving east a speed vo sees a red light in front of him and hits the brakes

after a short reaction. The car slows down at a rate of a1,2. A typical reaction time is 0.8

seconds. The situation is represented in the picture.

a) Where is the origin of the reference frame?

b) What information given in the problem is missing from the illustration? Add it to

the illustration.

c) What assumptions are made in Part I and Part II? How do these assumptions affect

the mathematical expressions that you can use in each part?



75

Homework


A stoplight turns yellow when you are 20 m from the edge of the intersection. Your car is

traveling at 12 m/s; after you hit the brakes, the car's speed decreases at a rate of 6.0 m/s

each second. (Ignore the reaction time needed to bring your foot from the floor to the brake

pedal.)

a) Sketch the situation. Decide where the origin of the coordinate system is and what

direction is positive.

b) Draw a motion diagram.

c) Draw an x(t) graph.

d) Draw a v(t) graph.

e) Write an expression for x(t) and v(t).

f) Use the expressions above to determine as many unknowns as you can.


A bus moving at 26 m/s (t = 0) slows at rate of 3.5 m/s each second. Sketch the situation.

Decide where the origin of the coordinate system is and what direction is positive.

a) Draw a motion diagram.

b) Use the expressions derived in this lesson and previous lessons to determine as

many unknowns as you can.

12.11 Reason and Represent

An object moves horizontally. The equations below represent its motion mathematically.

Describe the actual motion that these two equations together might describe.

a. v = +20 m/s + (−2 m/s2 )t

b. x = −200 m + (+20 m/s)t +1

2(−2 m/s2 )t 2

a) Describe the motion in words and sketch the process represented in the two

mathematical expressions above. Act it out.

b) Draw a motion diagram

c) Draw a position-versus-clock reading graph and a velocity-versus-clock reading

graph


d) Determine when and where the object will stop.


A remote control car runs down a driveway at an initial speed of 6.0 m/s for 8.0 sec, then

uniformly increases its speed to 9.75 m/s in 5.0 sec.

a) Sketch the situation, label all knowns and unknowns. Decide where the origin of the

coordinate system is and what direction is positive.

b) Draw a motion diagram.

c) Draw a v(t) graph.

d) Use the expressions in this and previous lessons to determine as many unknowns as

you can.

12.13 Regular problem

Examine the graph below.

a) Describe a real life situation that this graph could represent, be sure to include all the

information on the graph and any extra in your situation.

b) Determine two unknown physical quantities (one of them should be in the units of meters).

c) If the object was moving at a constant speed equal to the speed of the object on the graph at

t = 0, what would be the distance it traveled in 6 seconds? How does it compare to the

distance the object on the graph traveled in the same time interval? Does the answer make

sense to you?



77

Reflect: What did you learn in this lesson? How do you know how to

calculate the distance an accelerating object travels during some time

interval? What do you need to know to be able to find that distance?

78 PUM | Kinematics | Lesson 13: Experimental Design © Copyright 2013, June, Rutgers, The State University of New Jersey.

Lesson 13: Experimental Design


Design an experiment to test whether the fan cart moves at constant speed or constant

acceleration. Use available equipment.

a) Describe the experimental setup in words and with a picture.

b) List the quantities that you will measure and the quantities that you will calculate.

c) Make measurable predictions for the outcome of your experiment based on the two

different models of motion: constant speed (v = constant), constant acceleration (v =

v0 + at or v = at if v0 = 0 ).

How are the predictions different from the models?

d) Perform the experiment, record the results, and decide which model you could not

disprove and which model you were able to disprove.


Design an experiment to determine the acceleration of the fan cart, first using sugar packets

or any other mechanical things, and second using the motion detector. Compare the results.

Start with the experiment that does not involve the motion detector.

a) Describe the experimental setup in words and with a picture.

b) List the quantities that you will measure and the quantities that you will calculate.

c) Think about experimental uncertainties. Which instrument or which procedure will

give you the largest uncertainty?

d) Conduct the experiment and calculate the result. Express the value of the

acceleration within a range.

e) Then use the motion detector to measure the acceleration.

f) Compare the results and account for the differences. Think about the assumptions

that you made and the experimental uncertainties as a result of your equipment.

Homework

13.3 Communicating your results

Write a report for the experiments in 13.1 and 13.2. Discuss what you learned about the

acceleration of the fan cart.

Use the rubrics below.

PUM | Kinematics | Lesson 13: Experimental Design


79

Hypothesis-prediction-testing rubric (for 13.1) Scientific

Ability Missing An attempt

Needs some

improvement Acceptable

Is able to

distinguish

between a

hypothesis

and a

prediction.

No prediction is

made. The

experiment is not

treated as a

testing

experiment.


but it is identical to the

hypothesis.


and is distinct from

the hypothesis but

does not describe the

outcome of the

designed experiment.

A prediction is

made, is distinct

from the hypothesis,

and describes the

outcome of the

designed

experiment.

Is able to

make a

reasonable

prediction

based on a

hypothesis.

No attempt is

made to make a

prediction.


that is distinct from the

hypothesis but is not

based on it.


that follows from the

hypothesis but does

not have an if-and-

then structure.

A prediction is

made that is based

on the hypothesis

and has an if-and-

then structure.

Is able to

make a

reasonable

judgment

about the

hypothesis.

No judgment is

made about the

hypothesis.

A judgment is made

but is not consistent

with the outcome of

the experiment.

A judgment is made

and is consistent with

the outcome of the

experiment but

assumptions are not

taken into account.

A reasonable

judgment is made

and assumptions are

taken into account.

Rubric for 13.2 Scientific ability Missing An attempt Needs some

improvement

Acceptable

Is able to

evaluate the

results by means

of an

independent

method.

No attempt is

made to

evaluate the

consistency of

the results

using an

independent

method.

A second

independent method

is used to evaluate

the results.

However there is

little or no

discussion about the

differences in the

results due to the

two methods.

A second

independent

method is used to

evaluate the

results. The results

of the two methods

are compared

using experimental

uncertainties.

But there is little

or no discussion of

the possible

reasons for the

differences when

the results are

different.

A second

independent method

is used to evaluate

the results and the

evaluation is done

with the

experimental

uncertainties. The

discrepancy between

the results of the

two methods and

possible reasons for

the discrepancy are

discussed.

80 PUM | Kinematics | Lesson 14: Details of the Throw © Copyright 2013, June, Rutgers, The State University of New Jersey.

Lesson 14: Details of the Throw


Throw an object up so that it comes close to the ceiling but it does not hit it.

a) Describe the motion on the way up and on the way down.

b) Draw a vertical motion diagram for the motion on the way up

c) Draw a vertical motion diagram for the motion on the way down.

d) Are the changes in velocity arrows consistent?

e) Describe another real life situation that is similar to an object being thrown up into

the air.


The data recorded in the table at the right are a record for the up

and down motion of the center of a ball thrown upward (the y-

axis points up). Fill in the table that follows.

a) Draw a position-versus-clock reading graph

b) Draw a velocity-versus-clock reading graph. Find its

slope. What do you call this slope?

c) How does the slope on the position versus time graph relate to the points plotted on

the velocity versus time graph?

d) Use the velocity versus clock reading graph to determine the ball’s acceleration at

the very top of its trajectory. Is the change in velocity consistent throughout the

entire motion? On the way up? At the top? On the way down?

e) What is the ball’s velocity at the top?

f) Can you reconcile the answers to parts (d) and (e)? Explain.

g) If the ball had zero acceleration and zero velocity at the top of its motion, what

would happen to it?

Did you know?

The accepted value for the acceleration of objects close to Earth is 9.8 m/s2, commonly

referred to as "g". This was found experimentally in a manner close to the one you use in

the problem above.

Clock reading t (s)

Position y (m)

0.000 0.00

0.133 0.44

0.267 0.71

0.400 0.80

0.533 0.71

0.667 0.42

0.800 –0.04

PUM | Kinematics | Lesson 14: Details of the Throw


81

h) How does your value for the acceleration compare to the accepted value? What

might have caused any discrepancies?

14.3 Assumptions

You are jumping off a high dive into a pool of water. It took you 1.5 sec to hit the water.

a) Draw a motion diagram for your motion while you are in the air, ending several

seconds after you enter the water. Are the motions similar or different? Explain.

b) If you were to find the velocity with which you hit the water with, what assumptions

do you make about the motion as you hit the water?

c) If you dropped a water balloon from the football stadium grandstands which takes

1.8sec to hit the ground, what assumptions would you make about the final position

and the objects motion?

d) Discuss your answer with the class, are these valid assumptions, do they consider

the motion of the object falling or the motion of the object sitting on the ground.


A ball is hit by a baseball bat which is 1.2 m above the ground, straight up at 24.0 m/s.

a) Sketch the situation to the top of the path, be sure to label and list all important

givens and unknowns. State your assumptions.

b) Draw a motion diagram and describe what happens to the speed as the ball

approaches its maximum height.

c) Determine all possible unknowns for the object when it reaches maximum height.

14.5 Regular Problem

The fuel in a bottle rocket burns for 2.0 s. While burning, the rocket moves upward with an

acceleration of 30 m/s2.

a) What is the vertical distance that the rocket travels while the fuel is still burning, and

how fast is it traveling at the end of the burn?

b) After the fuel stops burning, the rocket continues upward but is now slowing at a

rate of about 10 m/s2. Estimate the maximum height that the rocket reaches. What

assumptions have you made in working through this problem?

Homework


A ball is thrown upward at 20 m/s from the top of a 150 m building.

82 PUM | Kinematics | Lesson 14: Details of the Throw © Copyright 2013, June, Rutgers, The State University of New Jersey.

a) Determine all the information about the ball when it is at a height of 165 m.

b) Determine all the information about the object when the ball just hits the ground.


James went outside and said he could throw a ball 25 meters upward. The ball takes 5.2

seconds to hit the ground.

Show that James could throw at least 25 m up. Discuss any assumptions you made to

determine this answer.

14.8 Reason

Examine the graph below.

a) Describe a real life situation that this graph could represent, be sure to include all the

information on the graph and any extra in your situation.

b) Determine the values of the physical quantities describing the motion during and at

the end of the time interval. Discuss assumptions you made.

14.9 Reason You hold two identical golf balls. You drop one ball and simultaneously throw

down the other ball. Explain what will be the same and different about their motion.

a) Represent the motion of the balls with two motion diagram next to each other.

b) Draw velocity versus time graphs for the two balls using the same axes.

c) Draw position versus time graphs using the same axes.

PUM | Kinematics | Lesson 15: Putting it all Together


83

Lesson 15: Putting it all Together

15.1 Derive (very challenging!)

So far we have been using two mathematical expressions when solving problems dealing

with the motion at constant acceleration.

v(t) = vo + a(t)

x(t) = xo + vo(t) + ½a(t)2

To find the displacement of an object, one must know the initial velocity, time interval and

acceleration. However, we often do not know how long the motion lasted, but we have

information about the final velocity (vf) instead of the time. How can we use the initial

velocity, the acceleration, and the final velocity of the object to find out how far it has

traveled?

Examine both equations above and think about how you can use the first one to derive the

expression for the time interval, which you can then substitute into the second. Show your

steps here.

Hint 1: Write an expression for t using the first equation.

Hint 2: Substitute the expression for t from the first equation into the second equation. This

way you will eliminate t from the second equation.

If you make no mistakes, you should get:

v2 = vo

2 + 2a(x-xo)

15.2 Practice

Use your new equation to solve the following problems:

a) You throw a marble up at the speed of 10 m/s. How high will it reach?

b) You drop a marble from a height of 1.5 m. What is the speed at which it will reach just

before it hits the ground?

c) A bus slows down from 15 m/s to 10 m/s in 200 meters. What is the acceleration of the

bus?

84 PUM | Kinematics | Lesson 15: Putting it all Together © Copyright 2013, June, Rutgers, The State University of New Jersey.

d) A bullet traveling at a speed of 200m/s passed through a wooden block and gets stuck in

it right as it is ready to come out. What is the acceleration of the bullet if the block is 40 cm

wide? (pay attention to units here)


Fill in the missing information on the graphs below.

Match the description with the vertical set of graphs.

a) An object decreasing speed then increasing speed in the opposite direction.

b) An object increasing speed

c) An object traveling at a constant velocity


Fill in the missing information on the graphs below. For each set of vertical graphs,

describe the motion for each segment.

Time (sec) P

osi

tio

n (

m)

Time (sec)

Vel

oci

ty (

m/s

)

Time (sec)

Acc

el.

Time (sec)

Po

siti

on

(m

)

Time (sec)

Vel

oci

ty (

m/s

)

Time (sec)

Acc

el.

Time (sec)

Po

siti

on

(m

)

Time (sec)

Vel

oci

ty (

m/s

)

Time (sec)

Acc

el.



85

15.5 Summarize

Analyze the information in the table below and complete the empty cells to summarize what

you know about motion with constant velocity and with constant acceleration using the

different representations of motion.

Describe the Motion… Motion at Constant Velocity Motion at Constant Acceleration.

In words and provide an

example.

The object's velocity is increasing by the

same amount every second.

For example, a cart going down a smooth

track that is tilted at an angle.

With a motion diagram.

With a graph of position

versus clock reading.

Mathematically as a function

x(t).

x(t) = xo + vo(t) + ½a(t)2


Describe the Motion… Motion with Constant Velocity Motion with Constant Acceleration.

With a graph of velocity



v(t). v(t) = constant

With a graph of acceleration



a(t).

a(t) = constant

Homework

15.6 Regular problem

A shuttle bus slows to a stop to avoid hitting a deer that had darted into the middle of the

road. If the bus was initially traveling at -13.7 m/s and had an average acceleration of + 4.1

m/s/s, what distance would it need to travel to avoid the collision?



87


Fill in the missing information on the graphs below. Then answer the questions that follow.

I II III

a) Which set(s) of the graphs represents the motion of an object thrown upwards?

b) Which set (s) of graphs represents the object moving in the negative direction?

c) Which set (s) of graphs represents the object first moving at constant speed and then

slowing down?

15.8 Evaluate the Solution

Identify any errors in the proposed solution to the following problem and provide a

corrected solution if there are errors.

The problem: A fire fighter slides down a fire pole at an increasing speed for 2.0 s, a

distance of 2.0 m (she holds on so she doesn’t move too fast at the bottom). She bends her

knees at the bottom and stops in 0.10 m.

a) Determine her speed at the end of the slide and just before she contacts the floor.

Time (sec)

Po

siti

on

(m

)

Time (sec)

Vel

oci

ty (

m/s

) Time (sec)

Acc

el.

Time (sec)

Po

siti

on

(m

)

Time (sec)

Vel

oci

ty (

m/s

)

Time (sec)

Acc

el.

Time (sec)

Po

siti

on

(m

)

Time (sec)

Vel

oci

ty (

m/s

)

Time (sec)

Acc

el.


b) What is her acceleration while stopping?

Proposed solution:

v = v0 + at = 0 + (9.8 m/s2)(2.0 s) = 19.6 m/s.

a = (v2

– v02) /2(x – x0) = [0

2 – (19.6 m/s)

2]/2(0.10 m) = –1920.8 m/s

2.


A beach ball is volleyed up from a height 2.1m above the ground into up to a height of 13m.

Find out the as many unknown physical variables about the initial conditions and 2.5

seconds later.


Heather and Komila are exercising in the park. When you start observing them, Komila is

50 m ahead of Heather. She is jogging at a speed of 5 mph and Heather is running at the

speed of 7 mph. How long will take Heather to catch up with Komila? What assumptions

did you make?


This time Heather and Komila are running towards each other. How long will it take them

to meet? What assumptions did you make?

15.12 Reason

Why is the head on collision of two cars more dangerous than the collision of cars traveling

in the same direction?

15.13 Pose a problem

Pose a problem about a real life situation involving motion. Decide what information you

can collect to solve it. Then use tools you learned in this module to solve it. Be prepared to

present your problem to class.

15.14 Reflect

Write a two-page summary of what you learned about motion so far.

Make sure that you focus on the most important ideas. Give specific

examples. The best summary will be used in the next edition of the module

to help students learn kinematics next year.

PUM | Kinematics | Lesson 16: Practice


89

Lesson 16: Practice


While concentrating on catching the football, a wide receiver on a football team runs into

the goal post. He was originally moving at 10 m/s and bounced back at 2.0 m/s. A video of

the collision indicates that it lasted 0.020 s. Determine the acceleration of the receiver

during the collision. Indicate any assumptions you made. How will you model the receiver

to solve the problem?


While traveling in your car at 24 m/s, you find that traffic has stopped 30 m in front of you.

Will you smash into the back of the car stopped in front of you? Your reaction time is 0.80 s

and the magnitude of your car’s acceleration is 8.0 m/s2 after the brakes have been applied.

List all assumptions you make.


Assume that the positive direction of the x-axis is to the right.

A car is moving according to the equation x = -30 (m) - 10 (m/s) t + 3(m/s2) t

2

a) Describe the motion of the car in words.

b) Determine the initial position, initial velocity, and acceleration of the car. Does it

speed up or slow down?

c) Draw a motion diagram for the car.

d) Draw a velocity versus time graph for the car. Write a function v(t) for the graph.

e) Sketch the position versus time graph right underneath the velocity versus time

graph. What do you expect to see on this graph at the instant when the car stops?

How far does it travel before it stops?

f) How long does it take for the car to stop? What happens after that if the acceleration

does not change? Is it a realistic situation?

90 PUM | Kinematics | Lesson 16: Practice © Copyright 2013, June, Rutgers, The State University of New Jersey.


The graph below represents the position versus time graph for two rockets: A and B.

See if anything massing on the graph. Add. Draw a picture of the motion of the two rockets.

a) Calculate the slope of each line. What is the speed of each rocket?

b) Write a mathematical expression x(t) for lines A and B.

c) How much distance was traveled by rocket A after 6 minutes?

d) How much time did it take rocket B to travel 60 km?

e) How much sooner did rocket B travel 60 km than rocket A?

f) If rocket A eventually landed at the same place it began, what would be the rocket’s

displacement for the entire trip?


a) You ride your bike west at a speed of 8.0 m/s. Your friend, 400 m east of you, is

riding her bike west at a speed of 12 m/s. (a) Fill in the table that follows. (Consider

the bikes as point-like objects.)

Sketch and translate:

Draw a sketch of the initial

situation and choose a

coordinate system to

describe the motion of

both bikes.

2 4 6 8 10 12 14 time

20

40

60

80

Rocket A

Rocket B



91

Represent physically:

Draw a motion diagram for

each bike.

Represent mathematically:

Construct equations that

describe the positions of

each bicycle as a function

of time x(t).

Solve and evaluate:

Use the equations to

determine when the

bicycles are at the same

position. Does your result

make intuitive sense?

b) After you fill in the table, draw on the left side below a position vs. time graph for

each bicycle using the same set of axes.

Position vs Time Graphs Velocity vs Time Graphs

c) Are the slopes of the two lines and their initial values consistent with the actual

motion and the coordinate system you used to describe the motion?

d) Does the graph appear to correspond with the calculated answer for the time when

the bicycles are at the same position? Explain.

e) Beside the above position-versus-time graph, draw a velocity-versus-time graph

representing the velocity of each bicycle on the same set of axes.

f) Are the signs consistent with the word description of the motion? Explain.




93

16.6 Real world application

The following chart is taken from the NJ MVC driver’s manual online1.

1609 m = 1.0 mi.

1.0 m = 3.28 ft

a) What is the speed, in mph, for the car traveling in the problem above (16.2)? Was

the answer close to the braking distance found in the NJ MVC driver’s manual? If

they are different, explain why.

b) Explain why the reaction distances increase as the speed increases?

c) Determine the reaction time they utilize in calculating the reaction distance in this

table? Why might the NJ MVC choose this time?

d) What is the car’s acceleration (in m/s2) for when it is traveling 70 mph? How does

this compare to the acceleration in 16.2?

16.7 Reason

a) Light from the Sun reaches Earth in about 8 minutes. If the speed of light is 3 x 108

m/s,

how far is the Sun? Can we consider it a point like object in this problem? Can we consider

Earth a point like object?

b) The next closest star to Earth is Proxima Centauris. It is 3.97 × 1013

km away from the

Solar system. If the star explodes tomorrow, when will see the explosion?

1 State of New Jersey Motor Vehicle Commission, http://www.state.nj.us/mvc, (August 2008)


16.8 Reflect

Imagine you could write a note to your past self about solving problems

related to motion. What would you say? Make sure that your instructions

are helpful for those who are starting the module.

16.9 Evaluate

Congratulations! You completed your first PUM module! What was

difficult about it? What was easy? What do you feel you excelled at? What

do you think you need to still work on? What can we do to improve the

module?

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