Chapter 4: Impulse and Momentum — Review and Exercises

Chapter contents: Chapter 4: Impulse and Momentum

Glossary

conservation of momentum principle

when the net external force is zero, the total momentum of the system is conserved or constant

elastic collision

a collision that also conserves total kinetic energy, in addition to the total momentum

impulse

the average net external force times the time it acts; equal to the change in momentum

inelastic collision

a collision in which total kinetic energy is not conserved

linear momentum

the product of mass and velocity

perfectly inelastic collision

a collision in which the colliding objects stick together

Section Summary

4.2: Linear Momentum and Force

Linear momentum (momentum for brevity) is defined as the product of a system’s mass multiplied by its velocity.
In symbols, linear momentum $p$ is defined to be
$p = m v$ ,
(4.2.1)
where $m$ is the mass of the system and $v$ is its velocity.
The SI unit for momentum is $kg \cdot m/s$ .
Newton’s second law of motion in terms of momentum states that the net external force equals the change in momentum of a system divided by the time over which it changes.
In symbols, Newton’s second law of motion is defined to be
$F_{net} = \frac{Δ p}{Δ t}$ ,
(4.2.2)
where $F_{net}$ is the net external force, $Δ p$ is the change in momentum, and $Δ t$ is the change time.

4.3: Impulse

Impulse is force times duration, and the net impulse, equal to the net external force multiplied by duration, gives the change in momentum:
$Δ p = F_{net} Δ t$ .
(4.3.1)
For non-constant net external force, you need to use time-averaged net external force to calculate the impulse over the duration.

4.4: Conservation of Momentum

The conservation of momentum principle says that the total momentum is conserved, or constant, in the absence of a net external force, as a direct consequence of Newton’s third law.
During projectile motion and where air resistance is negligible, momentum is conserved in the horizontal direction because horizontal forces are zero.

4.5: Elastic Collisions in One Dimension

An elastic collision is one that conserves total kinetic energy.
Conservation of kinetic energy and momentum together allow the final velocities to be calculated in terms of initial velocities and masses in one dimensional two-body collisions.

4.6: Inelastic Collisions in One Dimension

An inelastic collision is one in which the total kinetic energy changes (it is not conserved).
A collision in which the objects stick together is sometimes called perfectly inelastic because it reduces total kinetic energy as much as possible while conserving total momentum.

Questions and Exercises

4.2: Linear Momentum and Force

Exercise 1

A car travels around a circular track at constant speed. (a) Does its kinetic energy change? (b) Does its momentum change? (c) If the car doubles its speed, by what factors do its momentum magnitude and kinetic energy change? Explain.

Exercise 2

An object that has a small mass and an object that has a large mass have the same momentum. Which object has the largest kinetic energy?

Exercise 3

An object that has a small mass and an object that has a large mass have the same kinetic energy. Which mass has the largest momentum?

Exercise 4

A hockey puck slides with constant velocity on frictionless ice. It has nonzero momentum. Does maintaining that momentum require a net force? Explain using the relationship between net force and the rate of change of momentum.

Exercise 5

An empty cart and a loaded cart experience the same constant net force for the same time. (a) Compare their changes in momentum. (b) Compare their changes in velocity. Explain how your answers agree with both forms of Newton’s second law.

Exercise 6

A ball moves in a circle at constant speed. (a) Is its momentum constant? (b) Does a net force act on it? Explain using the relationship between force and momentum, and identify the direction of the net force.

Exercise 7

How can a small force impart the same momentum to an object as a large force?

Exercise 8

Football coaches advise players to block, hit, and tackle with their feet on the ground rather than by leaping through the air. Using the concepts of momentum, work, and energy, explain how a football player can be more effective with his feet on the ground.

4.3: Impulse

Exercise 9

Two carts of different masses start at rest on a level, frictionless track. Each is pushed with the same constant horizontal force. (a) If the forces act for the same time, do the carts gain the same momentum or the same kinetic energy? (b) In a separate trial, the carts again start at rest, but the forces act through the same distance. Do the carts gain the same momentum or the same kinetic energy? Explain.

Exercise 10

A 0.20-kg ball approaches a wall at 5.0 m/s. In one collision it stops; in another it rebounds at 5.0 m/s along the same line. (a) Find the magnitude and direction of the ball’s momentum change in each case. (b) If the collisions last equally long, which produces the greater average net force on the ball, and by what factor?

Exercise 11

Explain in terms of impulse how padding reduces forces in a collision. State this in terms of a real example, such as the advantages of a carpeted vs. tile floor for a day care center.

4.4: Conservation of Momentum

Exercise 12

Two skaters initially at rest push off each other and move in opposite directions. Each now has momentum. How can the total momentum of the two skaters remain zero?

Exercise 13

Under what circumstances is momentum conserved?

Exercise 14

Can momentum be conserved for a system if there are external forces acting on the system? If so, under what conditions? If not, why not?

Exercise 15

In projectile motion with negligible air resistance, horizontal momentum is conserved, but vertical momentum is not. (a) Explain this difference in terms of the forces acting on the projectile. (b) How does this illustrate the independence of perpendicular components of motion discussed in two-dimensional kinematics?

Exercise 16

Revisit the garden-hose scenario from Newton’s Third Law of Motion: Symmetry in Forces (Exercise 17). Explain the backward push on the hose in terms of momentum transferred to the water. How is this explanation connected to Newton’s third law?

Exercise 17

Explain in terms of momentum and Newton’s laws how a car’s air resistance is due in part to the fact that it pushes air in its direction of motion.

Exercise 18

An astronaut is initially at rest relative to a nearby spacecraft and is holding a tool. Neglect external forces during the following action. (a) In which direction should the astronaut throw the tool to move toward the spacecraft? (b) Explain why air is unnecessary for this method of propulsion. Use conservation of momentum in your explanation.

4.5: Elastic Collisions in One Dimension

Exercise 19

What physical quantities are conserved in an elastic collision?

Exercise 20

Give examples of elastic (or nearly elastic) collisions from everyday experiences.

Exercise 21

A cart moving at 4.0 m/s strikes an identical cart initially at rest on a level, frictionless track. Consider two proposed outcomes: A, the first cart stops and the second moves at 4.0 m/s; B, both carts move at 2.0 m/s in the original direction. (a) Which outcomes conserve momentum? (b) Which conserve kinetic energy? (c) Which could describe an elastic collision? Explain.

4.6: Inelastic Collisions in One Dimension

Exercise 22

What is an inelastic collision? What is a perfectly inelastic collision?

Exercise 23

A 0.50-kg cart moving at 4.0 m/s collides with an identical cart initially at rest on a level, frictionless track. They stick together. (a) Find their common final velocity. (b) Calculate the total kinetic energy before and after the collision. (c) Where can the missing kinetic energy go? (d) Why can the carts not both come to rest?

Exercise 24

A small pickup truck that has a camper shell slowly coasts toward a red light with negligible friction. Two dogs in the back of the truck are moving and making various inelastic collisions with each other and the walls. What is the effect of the dogs on the motion of the center of mass of the system (truck plus entire load)? What is their effect on the motion of the truck?

Source sections: Linear Momentum and Force, Impulse, Conservation of Momentum, Elastic Collisions in One Dimension, Inelastic Collisions in One Dimension. Credits and licenses appear with each source section.