# Vertical elastic collision with spring

Gothican
I don't usually ask questions here, but I've been stuck for a v-e-r-y long time on this problem.

It goes like this:
A ball is dropped 1 meter above a tray connected to a spring. This is what it looks like:

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|____|
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Mass of ball - 1kg
Mass of tray - 5kg
Spring - K = 100 N/m

If the collision between the ball and the tray is totally elastic, what is the maximum compression of the spring?

## Homework Equations

The velocity of the ball when it hits the tray is 4.47 m/s.

The probable equation would be something like this:
Up+Ek+Ee=Up+Ek+Ee
I just don't know what to put in.

I would really appreciate any help.
Thanks, Gothican

Edit: I'm using g=10

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RoyalCat
EDIT 3:
The following solution is invalid.

Use conservation of energy, and be smart about your choice of plane of reference.

Try looking at the energy of the system at the initial state, and at the final state.
If we choose the plane of reference at the height of the maximal compression of the spring:

Ei = Ug
Ef = Ue (Remember that the ball comes to a halt, so kinetic energy is eliminated, as is potential energy since the final state is at the plane of reference)

Try finding out what Ei and Ef are and plugging in all the data.

EDIT 1:
Whoops, didn't notice the tray had mass as well. In that case, you'll need to remember that the initial energy of the system includes the original compression of the spring, and the gravitational potential energy of the tray. You'll need to consider the forces acting on the tray to see what this original compression is.

EDIT 2:
My final answer was 0.551 meters, if you wish to compare.

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Gothican
Right, but what would be the difference between an elastic collision as in this case, and a plastic (inelastic) collision?

Oh, and your answer isn't right - it should come out at around 80 cm.

Mentor
If the collision is totally elastic, the ball bounces off the tray. Find the velocity of the tray just after the collision.

RoyalCat
Oh, I see, I assumed the collision was completely elastic. A case which also invalidates my assumption of conservation of energy, making my solution completely wrong.

I'm rather confused myself, now.

Gothican
Y-E-S!
Got it.

Thanks Doc; I assumed at the beginning that there should be some movement together because there was a spring, but come to think of it, there really shouldn't be.

Equation - Uelastic + Ukinetic +Ugravity = Uelastic

Gothican

RoyalCat
If the collision is totally elastic, the ball bounces off the tray. Find the velocity of the tray just after the collision.

Does the tray not move at all?

Gothican
It does - The ball hits the tray and then they each get different velocities figured out by the regular elastic collision equations:
V1 + U1 = V2 + U2
M1V1 + M2V2 = M2U2 + M1U1

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Mentor
Does the tray not move at all?
Sure it does. Find its post-collision speed using conservation of momentum and energy.