Chapter 7

Impulse and Momentum

Learning ObjectivesSystems of particles, linear momentum • Impulse and momentum

– Students should understand impulse and linear momentum, so they can:• Relate mass, velocity, and linear

momentum for a moving object, and calculate the total linear momentum of a system of objects.

• Relate impulse to the change in linear momentum and the average force acting on an object.

• Calculate the area under a force versus time graph and relate it to the change in momentum of an object.

Learning ObjectivesSystems of particles, linear momentum• Conservation of linear momentum, collisions

– Students should understand linear momentum conservation, so they can:• Identify situations in which linear momentum,

or a component of the linear momentum vector, is conserved.

• Apply linear momentum conservation to one-dimensional elastic and inelastic collisions and two-dimensional completely inelastic collisions.

• Analyze situations in which two or more objects are pushed apart by a spring or other agency, and calculate how much energy is released in such a process.

Table Of Contents

1. The Impulse-Momentum Theory

2. The Principle of Conservation of Linear

Momentum

3. Collisions in One Dimension

4. Collisions in Two Dimensions

5. Center of Mass

Chapter 7 Impulse and Momentum

Section 1:The Impulse-Momentum Theory

Which do you think has more momentum?

Momentum

• Momentum is a measure of how hard it is to stop or turn a moving object.

• What characteristics of an object would make it hard to stop or turn?– Mass– Velocity

Problem• Calculate the momentum of a 65-kg sprinter

running east at 10 m/s.

Problem• Calculate the momentum of a system composed of a 65-kg

sprinter running east at 10 m/s and a 75-kg sprinter running north at 9.5 m/s.

Change in momentum

• Like any change, change in momentum is calculated by looking at final and initial momentums.

• p = pf – pi

– p: change in momentum

– pf: final momentum

– pi: initial momentum

Momentum change demonstration• Using only a meter stick, find the momentum

change of each ball when it strikes the desk from a height of exactly one meter.

• Which ball, Bouncy or Lazy, has the greatest change in momentum?

Wording dilemma

• In which case is the magnitude of the momentum change greatest?

• In which case is the change in the magnitude of the momentum greatest?

Impulse (J)

• Impulse is the product of an external force and time, which results in a change in momentum of a particle or system.

• J = F t and J = P

• Therefore Ft = P• Units: N s or kg m/s (same as momentum)

Impulsive Forces• Usually high

magnitude, short duration.

• Suppose the ball hits the bat at 90 mph and leaves the bat at 90 mph, what is the magnitude of the momentum change?

• What is the change in the magnitude of the momentum?

Impulse (J) on a graph

F(N)

t (ms)0 1 2 3 40

1000

2000

3000

area under curve

Problem

• Suppose a 1.5-kg brick is dropped on a glass table top from a height of 20 cm.a) What is the magnitude and direction of the impulse

necessary to stop the brick?b) If the table top doesn’t shatter, and stops the brick in 0.01 s,

what is the average force it exerts on the brick?c) What is the average force that the brick exerts on the table

top during this period?

Problem

This force acts on a 1.2 kg object moving at 120.0 m/s. Thedirection of the force is aligned with the velocity. What is the new velocity of the object?

1,000

2,000

0.20 0.40 0.60 0.80 t(s)

F(N)

3,000

Chapter 7 Impulse and Momentum

Section 2:The Principle of Conservation of

Linear Momentum

Law of Conservation of Momentum

• If the resultant external force on a system is zero, then the vector sum of the momentums of the objects will remain constant.

• Pbefore = Pafter

Problem

• A 75-kg man sits in the back of a 120-kg canoe that is at rest in a still pond. If the man begins to move forward in the canoe at 0.50 m/s relative to the shore, what happens to the canoe?

External versus internal forces• External forces: forces coming from outside

the system of particles whose momentum is being considered.– External forces change the momentum of

the system.• Internal forces: forces arising from

interaction of particles within a system.– Internal forces cannot change momentum of

the system.

An external force in golf

• The club head exerts an external impulsive force on the ball and changes its momentum.

• The acceleration of the ball is greater because its mass is smaller.

The System

An internal force in pool

• The forces the balls exert on each other are internal and do not change the momentum of the system.

• Since the balls have equal masses, the magnitude of their accelerations is equal.

The System

Applying the Principle of Conservation of Linear Momentum

• Decide which objects are included in the system

• Relative to the system, identify the internal and external forces.

• Verify that the system is isolated• Set the final momentum of the system

equal to its initial momentum.– Remember that the momentum is a

vector

Explosions• When an object separates suddenly, as in an

explosion, all forces are internal.• Momentum is therefore conserved in an

explosion.• There is also an increase in kinetic energy in

an explosion. This comes from a potential energy decrease due to chemical combustion.

Recoil

• Guns and cannons “recoil” when fired.• This means the gun or cannon must move

backward as it propels the projectile forward.• The recoil is the result of action-reaction force

pairs, and is entirely due to internal forces. As the gases from the gunpowder explosion expand, they push the projectile forwards and the gun or cannon backwards.

Problem• Suppose a 5.0-kg projectile launcher shoots a

209 gram projectile at 350 m/s. What is the recoil velocity of the projectile launcher?

Problem• An exploding object breaks into three fragments. A 2.0 kg

fragment travels north at 200 m/s. A 4.0 kg fragment travels east at 100 m/s. The third fragment has mass 3.0 kg. What is the magnitude and direction of its velocity?

Chapter 7 Impulse and Momentum

Section 3:

Collisions in One Dimension

Collisions• When two moving objects make contact with

each other, they undergo a collision.• Conservation of momentum is used to analyze

all collisions.• Newton’s Third Law is also useful. It tells us

that the force exerted by body A on body B in a collision is equal and opposite to the force exerted on body B by body A.

Collisions• During a collision,

external forces are ignored.

• The time frame of the collision is very short.

• The forces are impulsive forces (high force, short duration).

Collision Types• Elastic collisions

– Also called “hard” collisions– No deformation occurs, no kinetic energy

lost• Inelastic collisions

– Deformation occurs, kinetic energy is lost• Perfectly Inelastic (stick together)

– Objects stick together and become one object

– Deformation occurs, kinetic energy is lost

(Perfectly) Inelastic Collisions• Simplest type of collisions.• After the collision, there is only one

velocity, since there is only one object.• Kinetic energy is lost.• Explosions are the reverse of perfectly

inelastic collisions in which kinetic energy is gained!

Problem• An 80-kg roller skating

grandma collides inelastically with a 40-kg kid. What is their velocity after the collision?

Problem• A fish moving at 2 m/s

swallows a stationary fish which is 1/3 its mass. What is the velocity of the big fish after dinner?

Elastic Collision• In elastic collisions, there is no deformation of

colliding objects, and no change in kinetic energy of the system. Therefore, two basic equations must hold for all elastic collisions

• pb = pa (momentum conservation)

• Kb = Ka (kinetic energy conservation)

Problem• A 500-g cart moving at 2.0 m/s on an air track elastically

strikes a 1,000-g cart at rest. What are the resulting velocities of the two carts?

Problem• Suppose three equally strong, equally massive

astronauts decide to play a game as follows: The first astronaut throws the second astronaut towards the third astronaut and the game begins. Describe the motion of the astronauts as the game proceeds. Assume each toss results from the same-sized "push." How long will the game last?

7.1.1. An astronaut drops a golf ball that is initially at rest from a cliff on the surface of the moon. The ball falls freely under the influence of gravity. Which one of the following statements is true concerning the ball as it falls? Neglect any frictional effects.

a) The ball will gain an equal amount of kinetic energy during each second.

b) The ball will gain an equal amount of momentum during each second.

c) The ball will gain an equal amount of momentum during each meter through which it falls.

d) The ball will gain twice the amount of speed for each meter through which it falls.

e) The amount of momentum the ball gains will be directly proportional to the amount of potential energy that it loses.

7.1.2. A football of mass m, initially at rest, is kicked so that leaves the foot at a speed v. If t represents the duration of the collision between the ball and the foot, which one of the following expressions determines the magnitude of the average force exerted on the ball?

a) mvt

b) (1/2)mv2

c) (1/2)mv2/t

d) mv/t

e) (1/2)mvt2

7.1.3. During the filming of a movie, a stunt person jumps from the roof of a tall building, but no injury occurs because the person lands on a large, air-filled bag. Which one of the following best describes why no injury occurs?

a) The bag increases the amount of time the force acts on the person and reduces the change in momentum.

b) The bag increases the amount of time during which the momentum is changing and reduces the average force on the person.

c) The bag decreases the amount of time during which the momentum is changing and reduces the average force on the person.

d) The bag provides the necessary force to stop the person.

e) The bag reduces the impulse to the person.

7.1.4. A sports car of mass m has the same kinetic energy as an SUV with a mass 3m as each is driven along the same road. Which vehicle, if either, has the largest momentum and what is the difference in their momenta, if any? Express your result as a percentage.

a) Since their kinetic energies are the same, their momenta are the same. The difference is zero percent.

b) The car has about 33% greater momentum than the SUV.

c) The car has about 42% greater momentum than the SUV.

d) The SUV has about 66% greater momentum than the car.

e) The SUV has about 73% greater momentum than the car.

7.1.5. Consider the following objects:(1) an electron (m = 9.1 × 1031 kg, v = 5.0 × 107 m/s)(2) the Hubble Space Telescope (m = 1.1 × 104 kg, v = 7.6 × 103 m/s)(3) a snail (m = 0.02 kg, v = 0.0003 m/s)(4) the largest super oil tanker (m = 1.5 × 108 kg, v = 2.0 m/s)(5) a falling rain drop (m = 0.0002 kg, v = 9.5 m/s)

Which one of these objects requires the greatest change in momentum to stop moving?

a) 1

b) 2

c) 3

d) 4

e) 5

7.1.6. A 28-kg beginning roller skater is standing in front of a wall. By pushing against the wall, she is propelled backward with a velocity of 1.2 m/s. While pushing, her hands are in contact with the wall for 0.80 s. Ignoring frictional effects, find the magnitude and direction of the impulse acting on the skater.

a) 34 Ns

b) +34 Ns

c) 42 Ns

d) +42 Ns

e) 53 Ns

7.1.7. A force that varies with time is applied to an object. The force increases linearly from zero to twenty-four newtons during the first two seconds. Then, the force remains constant for four seconds. During the last two seconds, the force linearly decreases to zero newtons. What is the total impulse exerted on the object?

a) This cannot be determined because neither the initial momentum nor the final momentum is known.

b) This cannot be determined because the mass of the object is not given.

c) The total impulse is 96 Ns.

d) The total impulse is 128 Ns.

e) The total impulse is 144 Ns.

7.1.8. A small asteroid collides with a planet. Which one of the following statements concerning what happens during the collision is correct?

a) The asteroid exerts a smaller force on the planet than the planet exerts on the asteroid.

b) The planet exerts a force on the asteroid, but the asteroid does not exert a force on the planet.

c) The asteroid exerts the same amount of force on the planet as the planet exerts on the asteroid.

d) The asteroid exerts a force on the planet, but the planet does not exert a force on the asteroid.

e) The planet exerts a smaller force on the asteroid than the asteroid exerts on the planet.

7.1.9. Three events are observed at a baseball game: I. A baseball is thrown by a pitcher. It starts from rest and is

traveling at +38 m/s as it flies toward the catcher. II. A baseball is traveling at +38 m/s when it enters the catcher’s

glove and stops. III. A baseball is traveling at +38 m/s when it hits a wall and

bounces away from the wall at 38 m/s.The change in the momentum of the baseball has the largest magnitude in which case(s)?

a) I only

b) II only

c) III only

d) I and II only

e) II and III only

7.1.10. A stone of mass m is dropped and falls freely under the influence of gravity. Ignoring any effects of air resistance, which one of the following expressions gives the momentum of the stone as a function of time t? The local acceleration due to gravity is g.

a)

b) gt

c)

d) mgt

e)

212 mgt

212 gt

mg

7.2.1. A 9-kg object is at rest. Suddenly, it explodes and breaks into two pieces. The mass of one piece is 6 kg and the other is a 3-kg piece. Which one of the following statements concerning these two pieces is correct?

a) The speed of the 6-kg piece will be one eighth that of the 3-kg piece.

b) The speed of the 3-kg piece will be one fourth that of the 6-kg piece.

c) The speed of the 6-kg piece will be one forth that of the 3-kg piece.

d) The speed of the 3-kg piece will be one half that of the 6-kg piece.

e) The speed of the 6-kg piece will be one half that of the 3-kg piece.

7.2.2. A sled of mass m is coasting at a constant velocity on the ice covered surface of a lake. Four birds, with a combined mass 0.5m, gently land at the same time on the sled. The sled and birds continue sliding along the original direction of motion. How does the kinetic energy of the sled and birds compare with the initial kinetic energy of the sled before the birds landed?

a) The final kinetic energy is one half of the initial kinetic energy.

b) The final kinetic energy is one third of the initial kinetic energy.

c) The final kinetic energy is one quarter of the initial kinetic energy.

d) The final kinetic energy is one ninth of the initial kinetic energy.

e) The final kinetic energy is equal to the initial kinetic energy.

7.2.3. Consider the recoil of a rifle when it is fired. Recoil is an example of which of the following concepts?

a) Newton’s Third Law of Motion

b) Impulse

c) Conservation of Momentum

d) all of the above

e) none of the above

7.2.4. Ryan, m = 45 kg, runs and jumps off a stationary, 168-kg floating platform on a lake. Ryan’s velocity as he leaps is +2.7 m/s. Ignoring any frictional effects, what is the recoil velocity of the platform?

a) 2.7 m/s

b) +0.72 m/s

c) 1.4 m/s

d) 0.72 m/s

e) +2.7 m/s

7.3.1. Object A is traveling due north while object B is traveling due south along the same line. The two objects have a head-on collision. Which one of the following statements best describes the velocities of the objects after the collision?

a) The final velocity of B will be the same as the initial velocity of A and vice versa.

b) The final velocities of both A and B will be less than the velocities before collision

c) The final velocities of A and B will be the same as they had before the collision.

d) The final velocities of A and B could be equal to zero m/s.

e) The final velocities of A and B will be equal to zero m/s.

7.3.2. A water balloon is dropped from a third story balcony. The balloon bursts when it hits the ground. It is a completely inelastic collision with the earth. How much kinetic energy is transferred to the earth as a result of the collision?

a) mv2/2

b) mv2

c) mv

d) 2mv2

e) zero joules

7.3.3. A ball is thrown downward; and its speed is v just before it strikes the ground. After the collision with the earth, it heads upward with a speed v. Which one of the following statements concerning this situation is true?

a) Only the momentum of the earth is changed by the collision.

b) The momentum of the earth is not changed by the collision.

c) Only the momentum of the ball is changed by the collision.

d) The momentum of the ball is not changed by the collision.

e) The collision causes both the momentum and the kinetic energy of the ball to change.

7.3.4. A rubber ball of mass m is dropped from rest from a window at a height h above the ground. Just before the ball hits the ground, it is traveling at a speed v. Just after it bounces, the ball is moving upward at a speed 0.75v. Which one of the following expressions gives the change in momentum of the ball as a result of the collision? Ignore any effects due to air resistance.

a)

b)

c)

d)

e)

24

mp gh

72

4

mp gh

7

4

mghp

24

mgp h

32

4

mgp h

7.3.5. In a scary science fiction movie, an asteroid is on a direct collision path with the Earth. The asteroid has mass m = 3.98 × 1015 kg and velocity v = 7.00 × 103 m/s. The Earth has mass M = 5.98 × 1024 kg and velocity V = +2.99 × 104 m/s. Assuming the collision is perfectly inelastic and that no mass is lost from the Earth/asteroid system, determine the velocity of the Earth after the collision.

a) +2.39 × 104 m/s

b) +2.48 × 104 m/s

c) +2.61 × 104 m/s

d) +2.82 × 104 m/s

e) +2.99 × 104 m/s

7.3.6. Two identical cars were involved in a collision at an icy intersection. Car A was stopped at a traffic light. Car B was moving at a speed v when it suffered a perfectly inelastic collision with the back end of car A. With what speed did the two cars slide into the intersection after the collision?

a) 2v

b) v

c) v/2

d) v/4

e) v/8

7.3.7. Two identical 2.0-kg objects are involved in a collision. The objects are on a frictionless track. The initial velocity of object A is +12 m/s and the initial velocity of object B is 6.0 m/s. What is the final velocity of the two objects?

a) Object A moves at 6.0 m/s and B moves at +12 m/s.

b) Object A moves at +9.0 m/s and B moves at +9.0 m/s.

c) Object A moves at 6.0 m/s and B moves at +6.0 m/s.

d) Object A moves at 12 m/s and B moves at +6.0 m/s.

e) This cannot be determined without more information.

7.3.8. A cart that can slide freely on a frictionless track has a vertical rigid sail as shown in the drawing and is at rest on the track. In case A, a ball of clay is launched horizontally as shown and strikes the sail, but does not stick. The ball falls straightdownward after the collision. In case B, a rubber ball, with the same mass and initialspeed as the clay ball, strikes the sail andbounces backward. Which one of the followingstatements concerning this situation is true?

a) The final velocity of the cart is the same in both cases and its magnitude is greater than zero m/s.

b) The final velocity of the cart is the larger in case A than it is in case B.

c) The final velocity of the cart is the smaller in case A than it is in case B.

d) The final velocity of the cart is zero m/s in both cases.

e) This question cannot be answered without knowing the masses of the balls and the cart and the initial speed of the balls.

Chapter 7 Impulse and Momentum

Section 4:

Collisions in Two Dimensions

2D-Collisions

• Momentum in the x-direction is conserved.

– Px (before) = Px (after)

• Momentum in the y-direction is conserved.

– Py (before) = Py (after)

• Treat x and y coordinates independently.

– Ignore x when calculating y

– Ignore y when calculating x

• Let’s look at a simulation:

– http://surendranath.tripod.com/Applets.html

Problem• Calculate velocity of 8-kg ball after the collision.

3 m/s

2 kg 8 kg

0 m/s

Before

2 m/s

2 kg

8 kgv

After

50o

x

y

x

y

Problem• A car with a mass of 950 kg and a speed of 16 m/s to the east

approaches an intersection. A 1300-kg minivan traveling north at 21 m/s approaches the same intersection. The vehicles collide and stick together. What is the resulting velocity of the vehicles after the collision?

Chapter 7 Impulse and Momentum

Section 5:Center of Mass

DEFINITION OF CENTER OF MASS

• The center of mass is a point that represents the average location for the total mass of a system

• The center of mass will remain constant during a collision

CENTER OF MASS

21

2211

mm

xmxmxcm

Velocity of the center of mass

21

2211

mm

vmvmvcm

21

2211

mm

xmxmxcm

Velocity of the center of mass• In an isolated

system, the total linear momentum does not change.

• Therefore, the velocity of the center of mass does not change.

21

2211

mm

vmvmvcm

7.4.1. On an air hockey table, two identical pucks have an elastic collision. Puck A is observed to be traveling northeast at v m/s before the collision and northwest at v m/s after the collision. Puck B is observed to be traveling northwest at v m/s before the collision and northeast at v m/s after the collision. Describe the direction of the impulse exerted on each disk during the collision.

a) The impulse on puck A is directed due west and the impulse on puck B is directed due east.

b) The impulse on puck A is directed due east and the impulse on puck B is directed due west.

c) The impulse on puck A is directed due north and the impulse on puck B is directed due north.

d) The impulse on puck A is directed northwest and the impulse on puck B is directed northeast.

e) The impulse on puck A is directed southwest and the impulse on puck B is directed southeast.

7.5.1. Consider the Earth and Mars in their orbit around the Sun. Where is the center of mass located for this three body system?

a) It is closer to the Earth, than it is to either the Sun or Mars.

b) It is at the center of a triangle that has the Sun at one apex, the Earth at another apex, and Mars at the third apex.

c) It is half of the distance between the Sun and Mars.

d) It is closer to the Sun, than it is to either the Earth or Mars.

e) It is closer to Mars, than it is to either the Earth or the Sun.

7.5.2. Two girl scouts are sitting in a large canoe on a still lake while at summer camp. The canoe happens to be oriented with the front of the canoe pointing due north. One of the girls is at the front of the canoe and the other is at the back of the canoe. The girl at the front walks to sit next to her friend at the back and sits down. What effect does this event have on the canoe?

a) The canoe will still be at rest, but it will be south of its original position.

b) The canoe will still be at rest, but it will be north of its original position.

c) The canoe will be moving toward the south.

d) The canoe will be moving toward the north.

e) The canoe will still be at rest at its original position.

7.5.3. A high school physics teacher also happens to be the junior hockey team coach. During a break at practice, the coach asks two players to go to the center of the ice with a 10.0-m pole. A 40-kg player is at one end of the pole and a 60-kg player is at the other end. The players then start pulling themselves together by pulling the rod and sliding on the ice as they move along the rod. When the two players meet, what distance will the 60-kg player have moved?

a) zero m

b) 4.0 m

c) 5.0 m

d) 6.0 m

e) 10.0 m

7.5.4. Two carts are placed on a horizontal air track. The mass of the first cart is m and the mass of the second cart is 1.5m. The first cart is accelerated to a speed v just before it collides with the second cart that is initially at rest. What is the speed of the center of mass of the system containing the two carts after the collision?

a) v

b) 4v/5

c) 2v/5

d) v/2

e) This cannot be determined without knowing if the collision is elastic or inelastic.

7.5.5. Children sometimes play with water rockets, in which an inner compartment is partially filled with water. The air pocket above the water is pressurized using a small pump. When the rocket is released from the pump, it flies upward at an initial velocity of 17 m/s as water exits through a nozzle at the bottom that has a radius of 2.5 mm. If the density of the water is 999 kg/m3, what is the initial thrust of the rocket?

a) 0.33 N

b) 5.7 N

c) 0.88 N

d) 2.4 N

e) 3.7 N