Kinematics – Part A

Physics 30S

Outcomes• S3P-3-02: Differentiate among position, displacement, and distance.• S3P-3-03: Differentiate between the terms “an instant” and “an interval” of

time.• S3P-3-04: Analyze the relationships among position, velocity, acceleration, and

time for an object that is accelerating at a constant rate. Include: transformations of position-time, velocity-time, and acceleration-time graphs using slopes and areas

• S3P-3-05: Compare and contrast average and instantaneous velocity for non-uniform motion. Include: slopes of chords and tangents

• S3P-3-06: Illustrate, using velocity-time graphs of uniformly accelerated motion, that average velocity can be represented as and that displacement can be calculated as

• S3P-3-07: Solve problems, using combined forms of:

Instant vs. Interval of Time

• What is an instant in time?– In physics, one point in time– For example, 3:15.42.35– One glance at the clock– Idealized concept

• An interval is a time period– For example, between 10:00 am and 11:00 am– 3:15.42.35 to 3:15.42.36

What Does it Mean on a Graph?

• Interval: section• Instant: single point• More to come later...

Homework

Journal entry:Compare and contrast an instant and an interval.

Use your ideas to guess what the difference is between instantaneous velocity and average velocity and between instantaneous acceleration and average acceleration.

INTRODUCTION TO KINEMATICS

The Meaning of Negatives in Kinematics

• Negatives indicate direction in kinematics!• Standard reference system – E/right is positive– W/left is negative– N/up is positive– S/down is negative

• Think of a number line!

SI Units• Système International des Unités (SI units)• Designed for consistency between all scientists• Always use SI units unless specifically instructed

otherwise!!!Quantity Unit

Displacement m

Velocity m/s

Acceleration m/s2

Time s

Mass kg

Force N

Position, Displacement and Distance

• Reminder:• Position: where are you right now• Distance: scalar, how far away are you from

the reference point• Displacement: vector, distance and a direction

Averages

• Averages intend to paint the “overall” picture• In physics, we will discuss average velocity and

average acceleration

Displacement, Velocity, and Acceleration - How Are They Related?

Remember, average velocity and acceleration are taken over an interval!Δ is the Greek letter delta. It means “change in”, as in Δt is change in time.

Problem solvingExample 1What is the average acceleration of a car which

speeds up to 120 km/h from 100.0 km/h in 1.5 seconds? Provide your answer in m/s2.

aav = 3.7 m/s2

Example 2What is the average velocity of a car which travels

25 m in 2.4 seconds? What is its velocity in km/h?Vav= 10. m/s

Working with Displacement, Velocity & Acceleration – Problem Solving

Example 3How far does a car travelling 100. km/h travel in

3.5 s? Provide your answer in m.d = 97 m

Example 4What is the new velocity of the car from question

3 if it accelerates at 2.2 m/s2 for 3.0 s?Vf = 34 m/s

Homework

• Physics, Concepts and Conceptions • Pg 30 #17, 19

DISPLACEMENT, VELOCITY AND ACCELERATION VS. TIME GRAPHS

Position, Velocity, Acceleration

• Which object is moving faster?

0 2 4 6 8 100

1

2

3

4

5

6

7

Distance vs. Time

Time (s)

Dist

ance

(m)

0 2 4 6 8 1002468

101214161820

Distance vs. Time

Time (s)

Dist

ance

(m

)

A B

Displacement Time Graphs

• We can graph the displacement travelled as a function of time

• Use this graph to tell us about displacement, velocity and acceleration

Constant Acceleration

• Many objects accelerate at a constant rate– Free fall (assume no air resistance)– Cart being pulled by an elastic

• Useful to understand what constant acceleration means in terms of velocity– Velocity is increasing by the same amount each

time interval– 5 km/h, 10 km/h, 15 km/h, 20 km/h, etc.

Constant Acceleration

0 2 4 6 8 100

20

40

60

80

100

120

Displacement vs. Time

Distance (m)

Time (s)

Disp

lace

men

t (m

)

0 2 4 6 8 100

0.5

1

1.5

2

2.5

Acceleration vs. Time

Acceleration (m/s2)

Time (s)

Acce

lera

tion

(m/s

2)

0 2 4 6 8 100

4

8

12

16

20

Velocity vs. Time

Velocity (m/s)

Time (s)

Velo

city

(m/s

)

Slope

• Slope is rise over run, the change in y over x• In a displacement time graph, y is

displacement and x is time– Remember velocity = displacement/time

• In a displacement time graph, the slope is velocity

• In a velocity time graph, the slope is acceleration

Constant Acceleration

0 2 4 6 8 100

20

40

60

80

100

120

Displacement vs. Time

Distance (m)

Time (s)

Disp

lace

men

t(m

)

0 2 4 6 8 100

0.5

1

1.5

2

2.5

Acceleration vs. Time

Acceleration (m/s2)

Time (s)

Acce

lera

tion

(m/s

2)

0 2 4 6 8 100

4

8

12

16

20

Velocity vs. Time

Velocity (m/s)

Time (s)

Velo

city

(m/s

)

Slope

Slope

Area

The area under a graph isA = lwA = xyFor a velocity time graph, y is velocity, x is timeA = vtBut d = vt, soThe area under a velocity time graph is the

displacement!

In general...

• The formula for the area under a graph will change depending on the shape of the graph

• We will always work with either constant velocity (square vt graph) or constant acceleration (triangle vt graph).

Constant Acceleration

0 2 4 6 8 100

20

40

60

80

100

120

Displacement vs. Time

Distance (m)

Time (s)

Disp

lace

men

t(m

)

0 2 4 6 8 100

0.5

1

1.5

2

2.5

Acceleration vs. Time

Acceleration (m/s2)

Time (s)

Acce

lera

tion

(m/s

2)

0 2 4 6 8 100

4

8

12

16

20

Velocity vs. Time

Velocity (m/s)

Time (s)

Velo

city

(m/s

)

Slope

Slope

Area

Area

Graph Questions – Example 1

1 2 3 4 5 6 7 802468

101214

Displacement

Time (s)

Dis

plac

emen

t (m

)

• Imagine the d t graph provides the displacement for a rollercoaster. To two sig figs, what is the velocity of the rollercoaster?

• V = 1.5 m/s

Graph Questions – Example 2

0.002.00

4.006.00

8.000.002.004.006.008.00

10.00

Velocity

Time (s)

Velo

city

(m/s

)• Imagine the v t

graph provides the velocity for a new rollercoaster dropping down the track. To three sig figs, what is the acceleration of the rollercoaster?

• a = 1.05 m/s2

Time (s)

Velocity (m/s)

1.00 1.052.00 2.103.00 3.154.00 4.205.00 5.256.00 6.307.00 7.358.00 8.40

Graph Questions – Example 3

0.002.00

4.006.00

8.000.002.004.006.008.00

10.00

Velocity

Time (s)

Velo

city

(m/s

)• Imagine the v t

graph provides the velocity for a rollercoaster dropping down the track. What is the displacement of the rollercoaster at 3.00s?

• d = 4.73m

Time (s)

Velocity (m/s)

1.00 1.052.00 2.103.00 3.154.00 4.205.00 5.256.00 6.307.00 7.358.00 8.40

Graph Questions – Example 4

0.001.00

2.003.00

4.005.00

6.007.00

8.000.00

0.50

1.00

1.50

2.00

2.50

Acceleration

Time (s)

Acc

cele

ratio

n(m

/s2)

• Imagine the a t graph provides the acceleration for a rollercoaster dropping down the track. What is the velocity of the rollercoaster at 6.00s?

• v = 13.2m/s

Graph Questions – Example 5

0.002.00

4.006.00

8.0010.00

12.00-8.00

-4.00

0.00

4.00

Displacement

Time (s)

Dis

plac

emen

t (m

)

• Imagine the d t graph provides the displacement for the rollercoaster’s track.

a)Explain what the rollercoaster is doing through each interval.

b) What is the velocity during each interval?

Time (s)

Displacement (m)

0.00 0.001.00 1.502.00 3.003.00 4.504.00 2.255.00 0.006.00 -2.257.00 -4.508.00 -6.759.00 -6.75

10.00 -6.7511.00 -3.3812.00 0.00

Graph Questions – Example 5 Solution

0.002.00

4.006.00

8.0010.00

12.00-8.00

-4.00

0.00

4.00

Displacement

Time (s)

Dis

plac

emen

t (m

)

a)The rollercoaster travels at 1.50 m/s for 3.0 s, changes direction and heads back towards the start, travelling at -2.25 m/s until 8.0s. The rollercoaster then stops until 9.0s, and then returns to the start at 3.38 m/s.

b) V1 = 1.50 m/s, V2 = -2.25 m/s, V3 = 0.00 m/s, V4 = 3.38 m/s

Homework

• Physics, Concepts and Conceptions • P.32 #28, 29, 30, 31• P.69 #14, 18, 20, 21

What about Instantaneous?• Imagine taking a smaller and

smaller interval:– 1 s, 0.1 s, 0.01s, 0.001s,

0.0001s, etc• Repeating this infinitely, there

would be no interval at all – you would have an instant

• The intersection points would merge together to create...

• A point of tangency!!!• Instantaneous velocity.ggb file

Tangent to the Curve

• The slope of the tangent line to the displacement graph at a given point in time is the instantaneous velocity at that given point in time

• Same relationship exists between velocity and instantaneous acceleration

Example 1

0.00 1.00 2.00 3.00 4.00 5.00 6.000.00

10.00

20.00

30.00

40.00

Velocity (m/s)

Time (s)

Velo

city

(m/s

)

Find the instantaneous acceleration at 2.00s.

Example 1

0.00 1.00 2.00 3.00 4.00 5.00 6.000.00

10.00

20.00

30.00

40.00

Velocity (m/s)

Time (s)

Velo

city

(m/s

)

Find the instantaneous acceleration at 4.00s.

Pt 1(2.00,0.00)Pt 2(6.50, 36.00)a = 8.00 m/s2

Preview to Calculus

• How do you tell when you’ve got the correct tangent line?

• Tangent lines are difficult to draw – do your best

• Tangent lines are like an estimation• To find the true instantaneous velocity or

acceleration, we need calculus!

Homework

• Physics, Concepts and Conceptions• P.34 # 36, 37, 38, 39, 43

Bringing Formulas into it

• S3P-3-06: Illustrate, using velocity-time graphs of uniformly accelerated motion, that average velocity can be represented as and that displacement can be calculated as

• Think back to the graph! The first formula is just slope!

• The second is just the first formula where we are solving for d!

Problem Solving

• Here are the formulas we have available:

• Pick one based on what you want to know and what is already known!

• (We are assuming constant acceleration for #2)

Example 1

• A puppy starts off at rest and accelerates at a constant rate to 8.0 m/s. Find his average velocity.

• 4.0 m/s

Example 2

• A puppy starts off at rest and accelerates at a constant rate. If his displacement is 250 m north after 40 s, find his average velocity.

• 6.25 m/s

Example 3

• A BMW stopped at a red light accelerates up to 110.0 km in 4.800 s. What is its average acceleration?

• 6.36 m/s2

Homework

• Basic Kinematics worksheet

Lab

The Plan

• Max Classes: 51. Time Intervals and Instants; Intro to

kinematics (displacement, velocity, acceleration)

2. dt, vt, at graphs3. Instantaneous acceleration4. Formulas and Problem solving5. Quiz