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Chapter 12: Momentum. 12.1 Momentum 12.2 Force is the Rate of Change of Momentum 12.3 Angular Momentum. Chapter 12 Objectives. Calculate the linear momentum of a moving object given the mass and velocity. Describe the relationship between linear momentum and force.

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chapter 12 momentum
Chapter 12: Momentum

12.1 Momentum

12.2 Force is the Rate of Change of Momentum

12.3 Angular Momentum

chapter 12 objectives
Chapter 12 Objectives

Calculate the linear momentum of a moving object given the mass and velocity.

Describe the relationship between linear momentum and force.

Solve a one-dimensional elastic collision problem using momentum conservation.

Describe the properties of angular momentum in a system—for instance, a bicycle.

Calculate the angular momentum of a rotating object with a simple shape.

chapter vocabulary
Chapter Vocabulary
  • angular momentum
  • collision
  • law of conservation of
  • momentum
  • elastic collision
  • gyroscope
  • impulse
  • inelastic collision
  • linear momentum
  • momentum
inv 12 1 momentum
Inv 12.1 Momentum

Investigation Key Question:

What are some useful properties of momentum?

12 1 momentum
12.1 Momentum
  • Momentum is a property of moving matter.
  • Momentum describes the tendency of objects to keep going in the same direction with the same speed.
  • Changes in momentum result from forces or create forces.
12 1 momentum1
12.1 Momentum
  • The momentum of a ball depends on its mass and velocity.
  • Ball B has more momentum than ball A.
12 1 momentum and inertia
12.1 Momentum and Inertia
  • Inertia is another property of mass that resists changes in velocity; however, inertia depends only on mass.
  • Inertia is a scalar quantity.
  • Momentum is a property of moving mass that resists changes in a moving object’s velocity.
  • Momentum is a vector quantity.
12 1 momentum2
12.1 Momentum
  • Ball A is 1 kg moving 1m/sec, ball B is 1kg at 3 m/sec.
  • A 1 N force is applied to deflect the motion of each ball.
  • What happens?
  • Does the force deflect both balls equally?
  • Ball B deflects much less than ball A when the same force is applied because ball B had a greater initial momentum.
12 1 kinetic energy and momentum
12.1 Kinetic Energy and Momentum
  • Kinetic energy and momentum are different quantities, even though both depend on mass and speed.
  • Kinetic energy is a scalar quantity.
  • Momentum is a vector, so it always depends on direction.

Two balls with the same mass and speed have the same kinetic energy but opposite momentum.

12 1 calculating momentum
p = m v12.1 Calculating Momentum
  • The momentum of a moving object is its mass multiplied by its velocity.
  • That means momentum increases with both mass and velocity.


(kg m/sec)

Velocity (m/sec)

Mass (kg)

comparing momentum
Comparing momentum

A car is traveling at a velocity of 13.5 m/sec (30 mph) north on a straight road. The mass of the car is 1,300 kg. A motorcycle passes the car at a speed of 30 m/sec (67 mph). The motorcycle (with rider) has a mass of 350 kg. Calculate and compare the momentum of the car and motorcycle.

  • You are asked for momentum.
  • You are given masses and velocities.
  • Use: p = m v
  • Solve for car: p = (1,300 kg) (13.5 m/s) = 17,550 kg m/s
  • Solve for cycle: p = (350 kg) (30 m/s) = 10,500 kg m/s
    • The car has more momentum even though it is going much slower.
12 1 conservation of momentum
12.1 Conservation of Momentum
  • The law of conservation of momentum states when a system of interacting objects is not influenced by outside forces (like friction), the total momentum of the system cannot change.

If you throw a rock forward from a skateboard, you will move backward in response.

12 1 collisions in one dimension
12.1 Collisions in One Dimension
  • A collisionoccurs when two or more objects hit each other.
  • During a collision, momentum is transferred from one object to another.
  • Collisions can be elastic


elastic collisions
Elastic collisions

Two 0.165 kg billiard balls roll toward each other and collide head-on.

Initially, the 5-ball has a velocity of 0.5 m/s.

The 10-ball has an initial velocity of -0.7 m/s.

The collision is elastic and the 10-ball rebounds with a velocity of 0.4 m/s, reversing its direction.

What is the velocity of the 5-ball after the collision?

elastic collisions1
Elastic collisions
  • You are asked for 10-ball’s velocity after collision.
  • You are given mass, initial velocities, 5-ball’s final velocity.
  • Diagram the motion, use m1v1 + m2v2 = m1v3 + m2v4
  • Solve for V3 : (0.165 kg)(0.5 m/s) + (0.165 kg) (-0.7 kg)=

(0.165 kg) v3 + (0.165 kg) (0.4 m/s)

  • V3 = -0.6 m/s
inelastic collisions
Inelastic collisions

A train car moving to the right at 10 m/s collides with a parked train car.

They stick together and roll along the track.

If the moving car has a mass of 8,000 kg and the parked car has a mass of 2,000 kg, what is their combined velocity after the collision?

  • You are asked for the final velocity.
  • You are given masses, and initial velocity of moving train car.
inelastic collisions1
Inelastic collisions
  • Diagram the problem, use m1v1 + m2v2 = (m1v1 +m2v2) v3
  • Solve for v3= (8,000 kg)(10 m/s) + (2,000 kg)(0 m/s)

(8,000 + 2,000 kg)

v3= 8 m/s

The train cars moving together to right at 8 m/s.

12 1 collisions in 2 and 3 dimensions
12.1 Collisions in 2 and 3 Dimensions
  • Most real-life collisions do not occur in one dimension.
  • In a two or three-dimensional collision, objects move at angles to each other before or after they collide.
  • In order to analyze two-dimensional collisions you need to look at each dimension separately.
  • Momentum is conserved separatelyin the xand ydirections.
chapter 12 momentum1
Chapter 12: Momentum

12.1 Momentum

12.2 Force is the Rate of Change of Momentum

12.3 Angular Momentum

12 2 force is the rate of change of momentum1
12.2 Force is the Rate of Change of Momentum
  • Momentum changes when a net force is applied.
  • The inverse is also true:
    • If momentum changes, forces are created.
  • If momentum changes quickly, large forces are involved.
12 2 force and momentum change
F = D p

D t

12.2 Force and Momentum Change

The relationship between force and motion follows directly from Newton's second law.

Force (N)

Change in momentum

(kg m/sec)

Change in time (sec)

calculating force
Calculating force
  • You are asked for force exerted on rocket.
  • You are given rate of fuel ejection and speed of rocket
  • Use F = Δ ÷Δt
  • Solve: Δ = (100 kg) (-25,000 kg m/s) ÷ (1s) = - 25,000 N
    • The fuel exerts and equal and opposite force on rocket of +25,000 N.

Starting at rest, an 1,800 kg rocket takes off, ejecting 100 kg of fuel per second out of its nozzle at a speed of 2,500 m/sec. Calculate the force on the rocket from the change in momentum of the fuel.

12 2 impulse
12.2 Impulse
  • The product of a force and the time the force acts is called the impulse.
  • Impulse is a way to measure a change in momentum because it is not always possible to calculate force and time individually since collisions happen so fast.
12 2 force and momentum change1
F D t = D p12.2 Force and Momentum Change

To find the impulse, you rearrange the momentum form of the second law.

Impulse (N•sec)

Change in



Impulse can be expressed in kg•m/sec (momentum units) or in N•sec.

chapter 12 momentum2
Chapter 12: Momentum

12.1 Momentum

12.2 Force is the Rate of Change of Momentum

12.3 Angular Momentum

inv 12 3 angular momentum
Investigation Key Question:

How does the first law apply to rotational motion?

Inv 12.3 Angular Momentum
12 3 angular momentum
12.3 Angular Momentum
  • Momentum resulting from an object moving in linear motion is called linear momentum.
  • Momentum resulting from the rotation (or spin) of an object is called angular momentum.
12 3 conservation of angular momentum
12.3 Conservation of Angular Momentum
  • Angular momentum is important because it obeys a conservation law, as does linear momentum.
  • The total angular momentum of a closed system stays the same.
12 3 calculating angular momentum
L = Iw12.3 Calculating angular momentum

Angular momentum is calculated in a similar way to linear momentum, except the mass and velocity are replaced by the moment of inertia and angular velocity.

Moment of inertia

(kg m2)



(kg m/sec2)




12 3 calculating angular momentum1
12.3 Calculating angular momentum
  • The moment of inertia of an object is the average of mass times radius squared for the whole object.
  • Since the radius is measured from the axis of rotation, the moment of inertia depends on the axis of rotation.
calculating angular momentum
Calculating angular momentum

An artist is making a moving metal sculpture. She takes two identical 1 kg metal bars and bends one into a hoop with a radius of 0.16 m. The hoop spins like a wheel. The other bar is left straight with a length of 1 meter. The straight bar spins around its center. Both have an angular velocity of 1 rad/sec. Calculate the angular momentum of each and decide which would be harder to stop.

  • You are asked for angular momentum.
  • You are given mass, shape, and angular velocity.
    • Hint: both rotate about y axis.
  • Use L= I, Ihoop = mr2, Ibar = 1/12 ml2
calculating angular momentum1
Calculating angular momentum
  • Solve hoop: Ihoop= (1 kg) (0.16 m)2 = 0.026 kg m2
    • Lhoop= (1 rad/s) (0.026 kg m2) = 0.026 kg m2/s
  • Solve bar: Ibar= (1/12)(1 kg) (1 m)2 = 0.083 kg m2
    • Lbar = (1 rad/s) (0.083 kg m2) = 0.083 kg m2/s
  • The bar has more than 3x the angular momentum of the hoop, so it is harder to stop.
12 3 gyroscopes angular momentum
12.3 Gyroscopes angular momentum
  • A gyroscope is a device that contains a spinning object with a lot of angular momentum.
  • Gyroscopes can do amazing tricks because they conserve angular momentum.
  • For example, a spinning gyroscope can easily balance on a pencil point.
12 3 gyroscopes angular momentum1
12.3 Gyroscopes angular momentum
  • A gyroscope on the space shuttle is mounted at the center of mass, allowing a computer to measure rotation of the spacecraft in three dimensions.
  • An on-board computer is able to accurately measure the rotation of the shuttle and maintain its orientation in space.
Jet Engines
  • Nearly all modern airplanes use jet propulsion to fly. Jet engines and rockets work because of conservation of linear momentum.
  • A rocket engine uses the same principles as a jet, except that in space, there is no oxygen.
  • Most rockets have to carry so much oxygen and fuel that the payload of people or satellites is usually less than 5 percent of the total mass of the rocket at launch.