Momentum

•  CAPA Assignment #7 is due today.

•  Covering material in Chapter 9 today 9.6-9.9

•  We will cover 9.1-9.5 next week.

Web page: http://www.colorado.edu/physics/phys1110/phys1110_sp12/

Announcements:

Momentum Defined momentum:

New version of Newton’s 2nd law:

Defined impulse:

Impulse–momentum theorem:

Momentum and kinetic energy Momentum and kinetic energy are both functions of the mass and velocity of an object. How do they differ?

Kinetic energy is a scalar, there is no directionality Momentum is a vector, has magnitude and direction

Kinetic energy comes from force applied over distance Momentum comes from force applied over time

Clicker question 1 Set frequency to BA

A.  Straight up ↑ B.  Straight down ↓ C.  Straight right → D.  Straight left ← E.  At an angle

A ball bounces off the floor as shown. The direction of the impulse of the ball is ...

Clicker question 1 Set frequency to BA

A.  Straight up ↑ B.  Straight down ↓ C.  Straight right → D.  Straight left ← E.  At an angle

A ball bounces off the floor as shown. The direction of the impulse of the ball is ...

The horizontal component of momentum (px) does not change, while the vertical component goes from down to up so the change is up

Vector subtraction:

Conservation of momentum Consider two objects, A and B, which interact with each other but there are no other forces acting

By Newton’s 3rd law, the force A exerts on B is equal and opposite to the force B exerts on A.

Since the forces are equal and opposite and the time over which the forces act must be equal, the impulse is also equal and opposite.

Therefore, the change in momentum of B by the force A exerts on B is equal and opposite to the change in momentum of A by the force B exerts on A

Conservation of momentum We can think of the two objects A and B as a system.

Forces between A and B can change the momentum of A and B but the net change of momentum of the system (due to these forces) is 0

Conservation of momentum: If there is no net force on a system, the total momentum is constant.

Clicker question 2 Set frequency to BA

A.  ¼ the velocity of the heavy cart B.  ½ the velocity of the heavy cart C.  equal to the velocity of the heavy cart D.  twice the velocity of the heavy cart E.  four times the velocity of the heavy cart

Consider two carts, of masses m and 2m, at rest on an air track and connected by a compressed spring. After releasing the two carts, the spring expands to its equilibrium point and drops off. The magnitude of the velocity of the lighter cart will be

Clicker question 2 Set frequency to BA

A.  ¼ the velocity of the heavy cart B.  ½ the velocity of the heavy cart C.  equal to the velocity of the heavy cart D.  twice the velocity of the heavy cart E.  four times the velocity of the heavy cart

Consider two carts, of masses m and 2m, at rest on an air track and connected by a compressed spring. After releasing the two carts, the spring expands to its equilibrium point and drops off. The magnitude of the velocity of the lighter cart will be

Consider the two carts as a system. They are initially at rest so the total momentum is 0. As long as no net external force acts it will stay 0. After the spring acts, the momenta must be equal and opposite. Since the velocity of the 2m cart must be ½ of the 1m cart.

Conservation of momentum Conservation of momentum can be extended to more than two objects.

Define a system of a number of objects and if no net force acts on the system then the total momentum remains constant

Conservation of momentum means that both the magnitude and direction of the total momentum remain constant.

Therefore, the individual components (x, y, z) must all individually be conserved.

Collisions Collisions are generally between two objects which have some initial momentum. After colliding the objects have some final momentum. If there are no other forces, momentum is conserved.

If the collision occurs over a short time, we can often ignore the effects of other forces

Clicker question 3 Set frequency to BA

A. 0 B. 1 C. 2 D. 4 E. 8

Ball A strikes a stationary ball B in a 1D collision. The initial momentum of ball A, , and the final momentum of ball B, , are shown on the graph. What is the magnitude of the x-component of the final ball A momentum ?

Clicker question 3 Set frequency to BA

A. 0 B. 1 C. 2 D. 4 E. 8

Ball A strikes a stationary ball B in a 1D collision. The initial momentum of ball A, , and the final momentum of ball B, , are shown on the graph. What is the magnitude of the x-component of the final ball A momentum ?

The initial total momentum of the system was . By momentum conservation the final total momentum must be the same so or

Definition of collisions Elastic collision: One in which kinetic energy is conserved. This is the case when conservative forces act during the collision

There are often small losses due to frictional forces that we usually ignore in order to see the big picture.

Examples of (approximately) elastic collisions collisions between billiard balls collisions between tennis ball and racket

Definition of collisions Inelastic collision: One in which kinetic energy is lost during the collision (non-conservative forces at work)

Generally true if objects are permanently deformed such as a car crash, or if there is significant heating

Completely inelastic collision: A collision in which both objects have the same final velocity (they move off together)

How to solve collision problems All collisions conserve momentum:

If problem is 2D or 3D, rewrite equation in each dimension

If the problem is 1D, just remove the arrows but remember the signs (positive & negative)!

Still need to remember to use correct signs

How to solve collision problems If we know the collision is elastic then we have another equation:

Since kinetic energy is a scalar, we only have one equation no matter how many dimensions are in the problem

If we know the collision is completely inelastic then we have a different additional equation

In components we get and

Example 1 Two air carts have equal mass. Cart A is moving toward cart B with speed vA1. Cart B is initially at rest. The carts lock together during collision. What is the speed of the carts after the collision? The carts are on a track and lock together. This is a completely inelastic collision in 1D which gives us two equations:

Let m=mA=mB and let v2=vA2=vB2 so

but vB1 is 0 so which gives us

Carts of unequal mass or where both are initially in motion will complicate the math but does not change the method

Example 2 Two air carts have equal mass and springs attached. Cart A is moving toward cart B with speed vA1. Cart B is initially at rest. What is the speed of the carts after the collision? The carts are on a track and have springs (conservative force). This is an elastic collision in 1D which gives us two equations:

Since masses are equal, can cancel them out. Also use vB1=0

Substituting gives

Either vA2 or vB2 is 0 and the other equal vA1. Physically, cart A must stop and cart B travels with velocity vA1.

Example 3 Newton’s Cradle. Ball A is moving toward Ball B with speed v. Ball A has twice the mass of Ball B and B is initially at rest. What is the speed of the balls after the collision?

As an exercise, I ask you to show that the mass of 2m

striking a ball of mass m (at rest) gives final velocities

Vm = +4/3v and V2m =+ v/3