Chapter 2

Motion in One Dimension

Group Question If I am traveling in a moving car and suddenly slam on my brakes, will I speed up or slow down? What quantity expresses this idea of change in speed? If I slam the car in reverse and floor it how do I differentiate moving backwards verses a forward speed?

History The first noted study of mechanics can be traced back to ancient Sumeria and Egypt whose people were primarily interested in the study of celestial bodies’ movements.

Geocentric Model The Greeks began in earnest their study of the skies around 300BC The geocentric model was accepted in this time which said the Earth was the center of the Universe

1609- Galileo brings physics into the modern era with his celestial observations, tracking the sun and our movement around it.

Dynamics The branch of physics involving the motion of an object and the relationship between that motion and other physics concepts

Kinematics is a part of dynamics In kinematics, you are interested in the description of motion

Not concerned with the cause of the motion

Quantities in Motion Any motion involves three concepts Displacement Velocity Acceleration

These concepts can be used to study objects in motion

Position Defined in terms of a frame of reference A choice of coordinate axes Defines a starting point for measuring the motion

Or any other quantity One dimensional, so generally the x- or y-axis

Displacement Defined as the change in position

f stands for final and i stands for initial

Units are meters (m) in SI

f ix x x

Displacement Examples From A to B

xi = 30 m xf = 52 m x = 22 m The displacement is

positive, indicating the motion was in the positive x direction

From C to F xi = 38 m xf = -53 m x = -91 m The displacement is

negative, indicating the motion was in the negative x direction

Displacement, Graphical

Exercise You start of from your house and travel 20.0m to the mailbox. What's the distance traveled? What's the displacement?

If you travel 15.0m to the tree first and then 15.0 to the mailbox what's your distance traveled? What's your displacement? At what angle did you leave the mailbox?

Vector and Scalar Quantities Vector quantities need both magnitude (size) and direction to completely describe them Generally denoted by boldfaced type and an arrow over the letter

+ or – sign is sufficient for this chapter

Scalar quantities are completely described by magnitude only

Displacement Isn’t Distance The displacement of an object is not the same as the distance it travels Example: Throw a ball straight up and then catch it at the same point you released it

The distance is twice the height The displacement is zero

ClassificationsScalars Vectors

Distance Displacement

Speed Velocity

Acceleration

Magnitude of a Force Forces

Magnitude of a Momentum

Momentum

Torque

Speed The average speed of an object is defined as the total distance traveled divided by the total time elapsed

Speed is a scalar quantity

total distanceAverage speed

total time

dv

t

Speed, cont Average speed totally ignores any variations in the object’s actual motion during the trip

The total distance and the total time are all that is important Both will be positive, so speed will be positive

SI units are m/s

Example I drive my car 0.50 km to the closest Chipotle. It takes me 2.5min to get there. I realize I forgot my wallet and have to drive back home to get it, then drive back which takes me 7.0min total (Ignoring the time it took to find the wallet). What is my average speed during the total trip?

Velocity It takes time for an object to undergo a displacement

The average velocity is rate at which the displacement occurs

Velocity can be positive or negative t is always positive

fi

averagefi

x xxv

t t t

Velocity continued Direction will be the same as the direction of the displacement, + or - is sufficient

Units of velocity are m/s (SI) Other units may be given in a problem, but generally will need to be converted to these

In other systems: US Customary: ft/s

Speed vs. Velocity Lets look at two cars going from point P to Q. Car Red travels in a straight line and Car Blue travels a swerved line. Both cars reach point Q at the same time. Which has a higher Velocity? Speed?

Speed vs. Velocity

Cars on both paths have the same average velocity since they had the same displacement in the same time interval

The car on the blue path will have a greater average speed since the distance it traveled is larger

Example 2.1 A turtle and a rabbit engage in a footrace over a distance of 4.00km. The rabbit runs 0.500km and then takes a nap for 90.0min. Upon awakening, he remembers the race and runs twice as fast. Finishing the course in a total time of 1.75hours the rabbit wins the race.

Example 2.1 Calculate the average speed of the rabbit.

What was his average speed before he stopped for a nap? Assume no detours or doubling back.

Graphical Interpretation of Velocity Velocity can be determined from a position-time graph

Average velocity equals the slope of the line joining the initial and final positions

An object moving with a constant velocity will have a graph that is a straight line

Average Velocity, Constant

The straight line indicates constant velocity

The slope of the line is the value of the average velocity

Notes on Slopes The general equation for the slope of any line is

The meaning of a specific slope will depend on the physical data being graphed

Slope carries units

change in vertical axisslope

change in horizontal axis

Average Velocity, Non Constant

The motion is non-constant velocity

The average velocity is the slope of the straight line joining the initial and final points

Instantaneous Velocity The limit of the average velocity as the time interval becomes infinitesimally short, or as the time interval approaches zero

The instantaneous velocity indicates what is happening at every point of time

lim0t

xv

t

Instantaneous Velocity on a Graph The slope of the line tangent to the position-vs.-time graph is defined to be the instantaneous velocity at that time The instantaneous speed is defined as the magnitude of the instantaneous velocity

Example 2.2 A train moves slowly along a straight portion of track according to the graph of position versus time. Find the average velocity for the trip, the average velocity during the first 4.00 seconds of motion, the average velocity during the second 4.00 seconds of motion and lastly the instantanious velocity at both t=2.00s and t=9.00s.

Example A motorist drives north for 35.0 minutes at 85.0 km/h and then stops for 15.0 minutes. He then continues north traveling 130km in 2 hours. What is his total displacement? What is his average velocity?

Acceleration Changing velocity means an acceleration is present

Acceleration is the rate of change of the velocity

Units are m/s² (SI), cm/s² (cgs), and ft/s² (US Cust)

fi

fi

v vva

t t t

Example A car is sitting on a hill. The E-Brake fails and the car starts rolling downhill. If after 10 seconds the car is moving 5m/s what is the acceleration?

Average Acceleration Vector quantity When the sign of the velocity and the acceleration are the same (either positive or negative), then the speed is increasing

When the sign of the velocity and the acceleration are in the opposite directions, the speed is decreasing

Negative Acceleration A negative acceleration does not necessarily mean the object is slowing down

If the acceleration and velocity are both negative, the object is speeding up

Instantaneous and Uniform Acceleration The limit of the average acceleration as the time interval goes to zero

When the instantaneous accelerations are always the same, the acceleration will be uniform The instantaneous accelerations will all be equal to the average acceleration

Graphical Interpretation of Acceleration Average acceleration is the slope of the line connecting the initial and final velocities on a velocity-time graph

Instantaneous acceleration is the slope of the tangent to the curve of the velocity-time graph

Average Acceleration

Relationship Between Acceleration and Velocity

Uniform velocity (shown by red arrows maintaining the same size)

Acceleration equals zero

Relationship Between Velocity and Acceleration

Velocity and acceleration are in the same direction Acceleration is uniform (blue arrows maintain the same length)

Velocity is increasing (red arrows are getting longer)

Positive velocity and positive acceleration

Relationship Between Velocity and Acceleration

Acceleration and velocity are in opposite directions

Acceleration is uniform (blue arrows maintain the same length)

Velocity is decreasing (red arrows are getting shorter)

Velocity is positive and acceleration is negative

Motion Diagram Summary

Please insert active figure 2.12

Kinematic Equations Used in situations with uniform acceleration

2

2 2

1212

2

o

o

o

o

v v at

x vt v v t

x v t at

v v a x

Notes on the equations

t2

vvtvx fo

av erage

Gives displacement as a function of velocity and time

Use when you don’t know and aren’t asked for the acceleration

Notes on the equations

Shows velocity as a function of acceleration and time

Use when you don’t know and aren’t asked to find the displacement

ov v a t

Notes on the equations

Gives displacement as a function of time, velocity and acceleration

Use when you don’t know and aren’t asked to find the final velocity

2o at

2

1tvx

Notes on the equations

Gives velocity as a function of acceleration and displacement

Use when you don’t know and aren’t asked for the time

2 2 2ov v a x

Example A race car starting from rest accelerates at a constant rate of 5.00m/s^2. What is the velocity of the car after it has travelled 30.5m? How much time has elapsed?

Example 2.5 A car traveling at a constant speed of 24.0m/s passes a sneaky trooper hidden behind a billboard. One second after the speeding car passes the billboard the trooper sets off in chase with a constant acceleration of 3.00m/s^2. How long does it take the trooper to overtake the speeding car? How fast is the trooper going?

Example A jet plane lands with a speed of 100m/s and can accelerate at a maximum rate of -5.00m/s^2 as it comes to rest.

What is the minimum time needed to come to a complete stop?

Is a 800m long runway going to be long enough to stop the plane?

Example A car is traveling due east at 25.0m/s at some instant. If its constant acceleration is 0.750m/s^2 due west, find its velocity after 8.50s have elapsed.

Free Fall All objects moving under the influence of gravity only are said to be in free fall Free fall does not depend on the object’s original motion

All objects falling near the earth’s surface fall with a constant acceleration

The acceleration is called the acceleration due to gravity, and indicated by g

Acceleration due to Gravity Symbolized by g g = 9.80 m/s²

When estimating, use g 10 m/s2

g is always directed downward Toward the center of the earth

Ignoring air resistance and assuming g doesn’t vary with altitude over short vertical distances, free fall is constantly accelerated motion

Free Fall – an object dropped Initial velocity is zero

Let up be positive

Use the kinematic equations Generally use y instead of x since vertical

Acceleration is g = -9.80 m/s2

vo= 0

a = g

Free Fall – an object thrown downward a = g = -9.80 m/s2

Initial velocity 0 With upward being positive, initial velocity will be negative

Free Fall -- object thrown upward Initial velocity is upward, so positive

The instantaneous velocity at the maximum height is zero

a = g = -9.80 m/s2 everywhere in the motion

v = 0

Thrown upward, cont. The motion may be symmetrical

Then tup = tdown

Then v = -vo

The motion may not be symmetrical Break the motion into various parts

Generally up and down

Non-symmetrical Free Fall

Need to divide the motion into segments

Possibilities include

Upward and downward portions

The symmetrical portion back to the release point and then the non-symmetrical portion

Example 2.8 A stone is thrown with a speed of 20.0m/s straight upward at a height of 50m above ground. Find:

Time it takes to reach max height

Max height Time needed to hit the ground Velocity and position at t=5.00s

Example It is possible to shoot an arrow at a speed as high as 100m/s.

If friction is neglected, how high would the arrow travel if shot straight up into the air?

How long would the arrow be in the air?