# Linear momentum and impulse problem

1. May 24, 2014

### BrainMan

1. The problem statement, all variables and given/known data
A 0.3-kg ball falls from a height of 30 m from rest and rebounds with half its velocity upon impact with a sidewalk. Find (a) the momentum delivered to the earth and (b) the impulse delivered to the earth.

2. Relevant equations
FΔt= mvf-mvi

3. The attempt at a solution
What I tried to do was I found the velocity using the potential energy formula to find the energy and then the kinetic energy formula to find the velocity right before the ball hit the ground. I then multiplied the velocity by the mass to find the initial momentum. After that I divided the velocity by two because it said that when it rebounded it had half the velocity. Then I multiplied half the initial velocity by the mass to get the final momentum. Then I used the equation FΔt= mvf-mvi to find the impulse. I got -3.7125 and the answer was 11 for both (a) and (b)

2. May 24, 2014

### Simon Bridge

Not too bad, but you missed out a bit.
1. Momentum is a vector (and it has units)
2. The collision involves the Earth - so you need to consider it in the calculation.
Law of conservation of momentum applies here.

Note: It is easier to just use momentum directly, so you avoid awkward numbers.
i.e. if velocity halves and mass stays the same then momentum halves.
if it ends up going in the opposite direction, and the initial momentum was positive, then the final momentum is negative.
$$K=\frac{mv^2}{2}=\frac{p^2}{2m}$$... since $v=p/m$
You want the gravitational PE lost falling turning into kinetic energy at the moment of contact so:$$\frac{p^2}{2m}=mgh \implies p=m\sqrt{2gh}$$... see?

3. May 24, 2014

### BrainMan

OK I see what I did wrong. I didn't make the velocity after the ball had bounced negative. I got the right numbers with my method but the negative threw off my answer. Thanks!

4. May 24, 2014

Well done :)