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## Homework Statement

An elastic object (F = k e) is thrown horizontally at a wall, so it is travelling at constant velocity, which then rebounds elastically (energy conserved). I'm only interested in when the object first makes contact and decelerates to 0ms-1 which will be when the object is at maximum compression. I thought horizontal motion would make it a bit simpler.

The object will come to rest compressed (before rebounding), due to the force of the collision. The deceleration it undergoes will be an average one, half the maximum which is at maximum compression. At rest/compression, all of the object's kinetic energy will be converted into elastic potential energy (assuming the object doesn't break).

I'd like to know the average, or maximum, force that the object experiences, due to the object's elasticity/spring constant, and due to the object's mass. I can then see quantitatively, for example, if a less-stiff (lower k) object absorbs more energy on impact, so the force that it experiences is less but extension greater (if this is the case).

## Homework Equations

F,max = - k x e

F,avg = m x a,avg

F,avg x t = -m x u

KE = 0.5 x m x v^2

PE = 0.5 x F x e

## The Attempt at a Solution

Not sure if it's best to go about it in terms of energy or momentum.

If I could work out the time it took to be compressed/stop, or the distance it took (i.e. the compression), I think it would help.

The following didn't get me far:

PE = 0.5 x F,max x e

F,max = 2PE / e

e = F,max / k

= 2PE / k e

e^2 = 2PE / k

PE = KE = 0.5 x m x u^2

e^2 = m x u^2 / k

e^2 / u^2 = m / k

e/u = sqrt( m/k )

t = sqrt( m/k) Is this anything?

Substituted t into

F,avg x t = -m x u

To get

F,avg^2 = k x m x u^2

But using that doesn’t agree with the equations of motion (v^2 = u^2 + 2as) I don't think.

Any advice or an alternative method would be good (preferably along the lines of the above).