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What is the difference between speed and velocity? Distance and Displacement?

What is the difference between distance and displacement?

What does direction have to do with anything?

Chapter 2 Kinematics in One

Dimension

Distance and Displacement

Kinematics is the branch of mechanics that describes the

motion of objects without necessarily discussing what

causes the motion.

Dynamics deals with the effect that forces have on motion.

Together, kinematics and dynamics form the branch of

physics known as Mechanics.

Kinematics

How far have you gone if you run around the track one time?

Distance vs Displacement

Distance ( d )• Total length of the path travelled• Measured in meters• scalar

Displacement ( )• Change in position (x) regardless of path

• x = xf – xi

• Measured in meters• vector

x

Distance vs Displacement

DistanceDisplacement

A

B

Chapter 2 Kinematics in One

Dimension

Speed and Velocity

The total distance traveled divided by the time required to cover the distance.

timeElapsed

Distance speed Average

SI units for speed: meters per second (m/s)

Average Speed

t

ds

How far does a jogger run in 1.5 hours (5400 s) if his average speed is 2.22 m/s?

m 12000s 5400sm 22.2d

Example 1: Distance Run by a Jogger

t

ds

Δtsd

The displacement divided by the elapsed time.

timeElapsed

ntDisplaceme velocityAverage

ttt o

o

xxx

v

Average Velocity

SI units for velocity: meters per second (m/s)

Andy Green in the car ThrustSSC set a world record of 341.1 m/s in 1997. To establish such a record, the driver makes two runs through the course, one in each direction, to nullify wind effects. From the data, determine the average velocity for each run.

Example 2 The World’s Fastest Jet-Engine Car

sm5.339s 4.740

m 1609

t

xv

sm7.342s 4.695

m 1609

t

xv

Example 2

What is the average velocity for both runs? (combined)

s341.1m

4.695)s (4.740

m 2*1609

t

ds

What is the average speed for both runs? (combined)

The instantaneous velocity indicates how fast the car moves and the direction of motion at each instant of time.

tt

xv

0lim

Instantaneous Velocity & Speed

The instantaneous speed is the magnitude of the instantaneous velocity

Chapter 2 Kinematics in One

Dimension

Acceleration

Describe the motion of the ball as it rolls down the ramp.

What happens to the displacement and velocity as time goes by?

Draw a Position vs time and Velocity vs Time graph of its motion

What if anything changes when the ramp is facing in the opposite direction?

ttt o

o

vvv

a

The difference between the final and initial velocity divided by the elapsed time

Acceleration

SI units for acceleration: meters per second per second (m/s2)

What does a negative acceleration mean?

Scenario Acceleration

Speeding up in the positive direction

Slowing down in the positive direction

Speeding up in the negative direction

Slowing down in the negative direction

Question #5A ball is thrown toward a wall, bounces, and returns to the thrower with the same speed as it had before it bounced. Which one of the following statements correctly describes this situation?

a) The ball was not accelerated during its contact with the wall because its speed remained constant.

b) The instantaneous velocity of the ball from the time it left the thrower’s hand was constant.

c) The only time that the ball had an acceleration was when the ball started from rest and left the hand of the thrower and again when the ball returned to the hand and was stopped.

d) During this situation, the ball was never accelerated.e) The ball was accelerated during its contact with the wall

because its direction changed.

Question #6

In an air race, two planes are traveling due east. Plane One has a larger acceleration than Plane Two has. Both accelerations are in the same direction. Which one of the following statements is true concerning this situation?

a) In the same time interval, the change in the velocity of Plane Two is greater than that of Plane One.

b) In the same time interval, the change in the velocity of Plane One is greater than that of Plane Two.

c) Within the time interval, the velocity of Plane Two remains greater than that of Plane One.

d) Within the time interval, the velocity of Plane One remains greater than that of Plane Two.

e) Too little information is given to compare the velocities of the planes or how the velocities are changing.

Question #7

Two cars travel along a level highway. An observer notices that the distance between the cars is increasing. Which one of the following statements concerning this situation is necessarily true?

a) Both cars could be accelerating at the same rate.

b) The leading car has the greater acceleration.

c) The trailing car has the smaller acceleration.

d) The velocity of each car is increasing.

e) At least one of the cars has a non-zero acceleration.

Question #8

The drawing shows the position of a rolling ball at one second intervals. Which one of the following phrases best describes the motion of this ball?

a) constant position

b) constant velocity

c) increasing velocity

d) increasing acceleration

e) decreasing velocity

Question #9

At one particular moment, a subway train is moving with a positive velocity and negative acceleration. Which of the following phrases best describes the motion of this train? Assume the front of the train is pointing in the positive x direction.

a) The train is moving forward as it slows down.

b) The train is moving in reverse as it slows down.

c) The train is moving faster as it moves forward.

d) The train is moving faster as it moves in reverse.

e) There is no way to determine whether the train is moving forward or in reverse.

Question #10

Which of the following velocity vs. time graphs represents an object with a negative constant acceleration?

Chapter 2 Kinematics in One

Dimension

Constant Acceleration Equations

1. a = acceleration

2. t = time (elapsed)

3. v = final velocity (at time t),

4. vo = initial velocity (at time 0)

5. x = position (at time t)

6. xo = initial position (at time 0)

AP Kinematic Variables:

t

vva o

ovvat

atvv o

Equations of Kinematics for Constant Acceleration

AP Equation#1

a

vvvvxx o

oo 21

t

vva o

a

vvt o

a

vvxx oo 2

22

Equations of Kinematics for Constant Acceleration

tvvxx 00 2

1

022 2 xxavv o

AP Equation#3

atvv o vvt

xx

0

0

2

1

221 attvxx oo

Equations of Kinematics for Constant Acceleration

vvv o 2

1

t

xxv 0

tvvxx 00 2

1

If, a is constant:

tatvvxx ooo 2

1

AP Equation#2

Question #11

Complete the following statement: For an object moving at constant, positive acceleration, the distance traveled

a) increases for each second that the object moves.

b) is the same regardless of the time that the object moves.

c) is the same for each second that the object moves.

d) cannot be determined, even if the elapsed time is known.

e) decreases for each second that the object moves

A spacecraft is traveling with a velocity of +3250 m/s. Suddenly the retrorockets are fired, and the spacecraft begins to slow down with an acceleration whose magnitude is 10.0 m/s2. What is the velocity of the spacecraft when the displacement of the craft is +215 km, relative to the point where the retrorockets began firing?

Example 8 An Accelerating Spacecraft

axvv o 222

d a v vo t+215,000 m -10.0 m/s2 ? +3250 m/s

axvv o 22

m 215000sm0.102sm3250 22 v

smv 2500 s

mv 2500

Question #12

An object moves horizontally with a constant acceleration. At time t = 0 s, the object is at x = 0 m. For which of the following combinations of initial velocity and acceleration will the object be at

x = 1.5 m at time t = 3 s?

a) v0 = +2 m/s, a = +2 m/s2 b) v0 = 2 m/s, a = +2

m/s2

c) v0 = +2 m/s, a = 2 m/s2 d) v0 = 2 m/s, a = 2

m/s2

e) v0 = +1 m/s, a = 1 m/s2

Question #13

An airplane starts from rest at the end of a runway and accelerates at a constant rate. In the first second, the airplane travels 1.11 m. What is the speed of the airplane at the end of the second second?

a) 1.11 m/s b) 2.22 m/s

c) 3.33 m/s d) 4.44 m/s

e) 5.55 m/s

Question #14

An airplane starts from rest at the end of a runway and accelerates at a constant rate. In the first second, the airplane travels 1.11 m. How much additional distance will the airplane travel during the second second of its motion?

a) 1.11 m b) 2.22 m

c) 3.33 m d) 4.44 m

e) 5.55 m

Chapter 2 Kinematics in One

Dimension

Section 6:

Freely Falling Bodies

In the absence of air resistance, it is found that all bodiesat the same location above the Earth fall vertically with the same acceleration. If the distance of the fall is smallcompared to the radius of the Earth, then the accelerationremains essentially constant throughout the descent.

This idealized motion is called free-fall and the accelerationof a freely falling body is called the acceleration due to gravity.

22

22

sft30sm10

sft2.32or sm80.9

org

g

Free Fall

Freefalling bodies

I could give a boring lecture on this and work through some examples, but I’d rather make it more real…

Use same kinematic equations just substitute g for aChoose +/- carefully to make problem as easy as possible

Free fall problems

The referee tosses the coin upwith an initial speed of 5.00m/s.In the absence if air resistance,how high does the coin go aboveits point of release?

Example 12

h a v vo t

? -9.80 m/s2 0 m/s +5.00 m/s

h a v vo t

? -9.80 m/s2 0 m/s +5.00 m/s

ahvv o 222 a

vvh o

2

22

m 28.1

sm80.92

sm00.5sm02

22

h

There are three parts to the motion of the coin. On the wayup, the coin has a vector velocity that is directed upward andhas decreasing magnitude. At the top of its path, the coin momentarily has zero velocity. On the way down, the coinhas downward-pointing velocity with an increasing magnitude.

In the absence of air resistance, does the acceleration of thecoin, like the velocity, change from one part to another?

Conceptual Example 14 Acceleration Versus Velocity

Does the pellet in part b strike the ground beneath the cliffwith a smaller, greater, or the same speed as the pelletin part a?

Conceptual Example 15 Taking Advantage of Symmetry

Question #19

Two identical ping-pong balls are selected for a physics demonstration. A tiny hole is drilled in one of the balls; and the ball is filled with water. The hole is sealed so that no water can escape. The two balls are then dropped from rest at the exact same time from the roof of a building. Assuming there is no wind, which one of the following statements is true?

a) The two balls reach the ground at the same time. b) The heavier ball reaches the ground a long time before the

lighter ball.c) The heavier ball reaches the ground just before the lighter ball.d) The heavier ball has a much larger velocity when it strikes the

ground than the light ball.e) The heavier ball has a slightly larger velocity when it strikes the

ground than the light ball.

Question #20Two identical ping-pong balls are selected for a physics demonstration. A tiny hole is drilled in one of the balls; and the ball is filled with water. The hole is sealed so that no water can escape. Each ball is shot horizontally from a gun with an initial velocity v0 from the top of a building. The following drawing shows several trajectories numbered 1 through 5. Which of the following statements is true?

a) Both balls would follow trajectory 5.

b) Both balls would follow trajectory 3.

c) The lighter ball would follow 4 and the heavier ball would follow 2.

d) The lighter ball would follow 4 and the heavier ball would follow 3.

e) The lighter ball would follow 4 or 3 and the heavier ball would follow 2

or 1, depending on the magnitude of v0.

Question #21

A cannon directed straight upward launches a ball with an initial speed v. The ball reaches a maximum height h in a time t. Then, the same cannon is used to launch a second ball straight upward at a speed 2v. In terms of h and t, what is the maximum height the second ball reaches and how long does it take to reach that height? Ignore any effects of air resistance.

a) 2h, t b) 4h, 2t

c) 2h, 4t d) 2h, 2t

e) h, t

Chapter 2 Kinematics in One

Dimension

Section 7:

Graphical Analysis of Velocity and Acceleration

Calculus – the abridged addition

Displacement Velocity acceleration

Slope of the line(derivative)

Area under the curve(integral)

sm4s 2

m 8 Slope

t

x

Finding velocity

Instantaneous Velocity

2sm6s 2

sm 12 Slope

t

v

Finding acceleration

Finding displacement

hbwlarea 2

1

velo

city

time

vo

t

v

v – vo = at

atttvd o 2

1

20 2

1attvd

Question #22A dog is initially walking due east. He stops, noticing a cat behind him. He runs due west and stops when the cat disappears into some bushes. He starts walking due east again. Then, a motorcycle passes him and he runs due east after it. The dog gets tired and stops running. Which of the following graphs correctly represent the position versus time of the dog?

Question #23

The graph above represents the speed of a car traveling due east for a portion of its travel along a horizontal road. Which of the following statements concerning this graph is true?

a) The car initially increases its speed, but then the speed decreases at a constant rate until the car stops.b) The speed of the car is initially constant, but then it has a variable positive acceleration before it stops.c) The car initially has a positive acceleration, but then it has a

variable negative acceleration before it stops.d) The car initially has a positive acceleration, but then it has a

variable positive acceleration before it stops.e) No information about the acceleration of the car can be

determined from this graph.

Question #24

Consider the position versus time graph shown. Which curve on the graph best represents a constantly accelerating car?

a) A

b) B

c) C

d) D

e) None of the curves represent a constantly accelerating car.

Question #25Consider the position versus time graph shown. Which curve on

the graph best represents a car that is initially moving in one direction and then reverses directions?

a) A

b) B

c) C

d) D

e) None of the curves represent a car moving in one direction then reversing its direction.