Newtonian Mechanics

Chapter 2

One Dimensional Motion

Learning ObjectivesNewtonian Mechanics-Kinematics

I.A.1. Motion in one dimensiona) Students should understand the general relationships among

position, velocity, and acceleration for the motion of a particle along a straight line, so that:1) Given a graph of one of the kinematic quantities, position,

velocity, or acceleration, as a function of time, they can recognize in what time intervals the other two are positive, negative, or zero, and can identify or sketch a graph of each as a function of time.

b) Students should understand the special case of motion with constant acceleration, so they can:1) Write down expressions for velocity and position as functions

of time, and identify or sketch graphs of these quantities.2) Use the equations, v = vo + at, x = xo + vot + (1/2)at2, and

v2 = vo2 + 2a(x – xo) to solve problems involving one-

dimensional motion with constant acceleration.

Table of Contents

2.1 Displacement

2.2 Speed and Displacement

2.3 Acceleration

2.4 Kinematic Equations

2.5 Applications of Kinematic Equations

2.6 Free Falling Bodies

2.7 Graphical Analysis of Kinematics

Chapter 2 Kinematics in One

Dimension

Section 1:

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

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

d

Distance vs Displacement

DistanceDisplacement

A

B

Question #1

In the morning, a bird is in Tampa, Florida. In the afternoon, the bird is near Orlando, Florida. Given this information, which one of the following statements best describes the relationship between the magnitude of the bird’s displacement and the distance the bird traveled?

a) The distance traveled is either greater than or equal to the magnitude of bird’s displacement.

b) The distance traveled is either less than or equal to the magnitude of bird’s displacement.

c) The distance traveled is equal to the magnitude of bird’s displacement.

d) The distance traveled is either less than or greater than the magnitude of bird’s displacement.

e) The distance traveled is greater than the magnitude of bird’s displacement.

Chapter 2 Kinematics in One

Dimension

Section 2:

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

dv

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

dv

Δtvd

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

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

Question #2

A race car, traveling at constant speed, makes one lap around a circular track of radius r in a time t. The circumference of a circle is given by C = 2r. Which one of the following statements concerning this car is true?

a) The displacement of the car does not change with time.b) The instantaneous velocity of the car is constant.c) The average speed of the car is the same over any time

interval.d) The average velocity of the car is the same over any time

interval.e) The average speed of the car over any time interval is equal

to the magnitude of the average velocity over the same time interval.

Question #3

A turtle and a rabbit are to have a race. The turtle’s average speed is 0.9 m/s. The rabbit’s average speed is 9 m/s. The distance from the starting line to the finish line is 1500 m. The rabbit decides to let the turtle run before he starts running to give the turtle a head start. What, approximately, is the maximum time the rabbit can wait before starting to run and still win the race?

a) 15 minutes b) 18 minutes

c) 20 minutes d) 22 minutes

e) 25 minutes

Question #4

A turtle, moving at a constant velocity of 0.9 m/s due south, is in a race with a rabbit, who runs at a moderate speed of 9 m/s. When the turtle is 45 m from the finish line, the rabbit begins taunting the turtle by running from the turtle to the finish line (without crossing it) and back to the turtle. The rabbit continues going back and forth between the turtle and the finish line until the turtle crosses the finish line. About how many meters does the rabbit travel as the turtle travels that last 45 m? Assume the rabbit always runs at 9 m/s and doesn’t lose any time changing direction.

a) 180 m b) 270 m

c) 360 m d) 450 m

e) 540 m

Chapter 2 Kinematics in One

Dimension

Section 3:

Acceleration

The notion of acceleration emerges when a change in velocity is combined with the time during which the change occurs.

Acceleration

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)

Determine the average acceleration of the plane.

sm0ov

s 0ot

hkm260v

s 29t

s

hkm0.9

s 0s 29

hkm0hkm260

a

Example 3Acceleration and Increasing Velocity

o

o

tt

vv

a

s 9s 12

sm28sm13

o

o

tt

vva

Example 4 Acceleration and Decreasing Velocity

2sm0.5a

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

A police cruiser is parked by the side of the road when a speeding car passes. The cruiser follows the speeding car. Consider the following diagrams where the dots represent the cruiser’s position at 0.5-s intervals. Which diagram(s) are possible representations of the cruiser’s motion?

a) A only

b) B, D, or E only

c) C only

d) E only

e) A or C only

Question #10

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

Chapter 2 Kinematics in One

Dimension

Section 4:

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:

It is common to dispense with the use of boldface symbols overdrawn with arrows for the displacement, velocity, and acceleration vectors (AP does not show arrows on given equations nor expect them on open-ended problems). We will, however, continue to convey the directions with a plus or minus sign. (AP calls elapsed time “t” where t = t – to)

t

vva o

Equations of Kinematics for Constant Acceleration

o

o

tt

vv

a

t

vva o

ovvat

atvv o

Equations of Kinematics for Constant Acceleration

AP Equation#1

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

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

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

Question #12

Complete the following statement: For an object moving with a negative velocity and a 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.

Question #13

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.

Chapter 2 Kinematics in One

Dimension

Section 5:

Applications of the Kinematic Equations

1. Make a drawing.

2. Decide which directions are to be called positive (+) and

negative (-).

3. Write down the values that are given for any of the

kinematic variables.

4. Write the appropriate equation(s).

5. Simplify/Rearrange equation(s) to be explicit for needed

solution

6. Substitute values (w/ units) into equation.

7. Solve and write solution (w/ units)

AP Reasoning Strategy

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 #14

Starting from rest, two objects accelerate with the same constant acceleration. Object A accelerates for three times as much time as object B. Which one of the following statements is true concerning these objects at the end of their respective periods of acceleration?

a) Object A will travel three times as far as object B.

b) Object A will travel nine times as far as object B.

c) Object A will travel eight times as far as object B.

d) Object A will be moving 1.5 times faster than object B.

e) Object A will be moving nine times faster than object B.

Question #15

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 #16

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 #17

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

Question #18

A passenger train starts from rest and leaves a station with a constant acceleration. During a certain time interval, the displacement of the train increases to three times the value it had at the start of that interval. During that same time interval, determine the increase in the train’s velocity. Let v represent the speed of the train at the end of the time interval; and v0 represent the speed at the beginning of the interval.

a) v = v0 b) v = 1.4v0

c) v = 1.7v0 d) v = 2.0v0

e) v = 3.0v0

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)

sm4

s 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.