Momentum & Collisions

Change Object’s Momentum

A net force must act on it during some time

Δp = F^net (Δt)

Equal to the area under the graph’s curve

Collisions

• Typically a large force over a small time

• Total momentum is conserved

• Impulse determines the velocity change

Momentum Conservation

When two bodies collide and the only forces involved are between them, their total momentum is conserved

A parent and child are initially at rest together on frictionless ice. They push off from each other, and the parent ends up moving to the right at 5 m/s. If the child is half the weight of the parent, with what speed does the child end moving?

10 m/s

They start off with no kinetic energy and they end up with nonzero kinetic energy

You are on a cart initially at rest that rides on a frictionless track. If you throw a ball off the cart towards the left, will the cart be put into motion?

Yes, to the right.

Momentum of the ball moves to the left, so the momentum of the cart has to move to the right because momentum is conserved so it has to balance it out to make it equate to zero

Same cart, initially at rest. You throw the ball at a vertical surface that is firmly attached to the cart. If the ball bounces straight back as shown, will the cart be in motion after the ball bounces back?

Yes, to the left

The ball has momentum to the right, so the cart has to have negative direction momentum to the left

Same cart, initially at rest. You throw a ball at a vertical surface and the ball stuck to the wall. Will the car be in motion after the ball bounces back?

No, the cart wouldn’t move

• Initially: cart and ball are both at rest, so momentum is zero

• After it sticks: the cart and ball are now one system, and momentum must be conserved so it has to equal to zero. Therefore, the cart will not move

Same cart, but moving at constant speed v to the left. If you throw a massive ball straight up (from your perspective), how will the speed of the cart change?

Will not change because momentum conservation can be separated into x and y directions

Elastic Collisions

Kinetic energy is conserved, more likely to hard undeformable objects like billiard balls

Inelastic Collisions

Some energy transformed into another form (noise, heat, deformation, internal friction, etc.)

Perfectly Inelastic Collisions

When objects stick together and can then be treated as a single mass

A constant force is exerted on a cart that is initially at rest on a frictionless air track. The force acts for a short time interval and gives the cart its final speed. To reach the same speed using a force that is half as big, the force must be exerted for a time interval that is

Twice as long

Impulse v.s Momentum v.s Speed

Impulse determines the change in momentum, which in turn determines the speed

Consider two objects A and B, with masses m_A > m_B that bounce off each other as illustrated below. What can you say about the change in momentum that they undergo?

Change in A’s momentum = - Change in B’s momentum

Due to Newton’s Third Law

Conservation of Momentum in an Isolated System

All impulses are imparted pairwise with Newton’s third law force pairs

Total Impulse on All Objects = 0

Total Momentum Change = 0

Total Momentum Before = Total Momentum After

This is true even if there are dissipative/friction forces between the particles, as long as there are no external forces

You have a choice of throwing a 100 g hard rubber ball or a 100 g lump of soft sticky clay at a target. Your goal is to knock the target over. If the two options are the same size, shape, etc, such that you could throw them at the same speed, which one would you use?

Hard rubber ball because it is likely to bounce off the sign nearly elastically, which would give you double the amount of impulse compared to the sticky ball

A fireworks projectile is travelling upwards (as shown in the figure) just before it explodes. Sets of possible momentum vectors for the fragments after the explosion are shown. Which sets could actually occur?

3 and 4 (in the same general direction)

Total Momentum After Explosion

Must be in the same direction as the initial velocity

Centre of Mass

The weighted average of the system’s position

Approximately where is the centre of mass of the earth-moon system?

Within the radius of earth, but not at the centre

Why does a high jumper do a back-bend over the bar?

It raises the height they can clear relative to their centre of mass

Can CM trajectory be changed?

Nothing the jumper does during flight can alter the CM trajectory besides external forces like gravity

A bullet is shot into a wood block attached to a pivot with massless strings. The collision happens so rapidly that the block doesn’t begin to swing upward until after the bullet is stuck. The momentum of the bullet plus block remains constant when:

• The bullet approaches (there are no external forces in the horizontal direction)

• The bullet stops (there are no external forces in the horizontal direction)

• Not when the block swings upwards (as the block swings upwards, tension in the ropes exert a net external force so total momentum is no longer conserved)

Two railroad cars are each the same mass m. Car A, traveling at velocity v_i, strikes B (initially at rest) and they lock together. How much kinetic energy is lost in the collision?

Half the initial kinetic energy

Identical constant forces continuously push identical Blocks A and B from the start line to the finish line. Block A is initially at rest and Block B is initially moving right. Which block has the greater change in momentum while moving from the start to the finish line?

A (because the change in velocity is greater)

Both blocks A and B start at rest, but A has 4 times as much mass as block B. The same constant force acts on them both. Which block incurs the greater change in momentum while moving from the start to the finish line?

A, because mass is greater (proportional to impulse)