Finding Maximum Spring Compression in Projectile Collision with Target

[Fernando PJ]

Homework Statement

A projectile of mass m is shot (with velocity v) at target of mass M which has a hole containing a spring inside (with constant spring constant K), and the projectile hits the spring. The target is initially at rest, and can slide without friction on a horizontal surface (see the figure). Find the distance Δx that the spring compress at maximum.
https://drive.google.com/file/d/0BySX8CbZB9eXR0RCLXRLcDl1eHM/view?usp=sharing Please find the figure attached.

Homework Equations

Potential elastic energy
U=1/2kx²
Kinetic energy
K=1/2mv²

The Attempt at a Solution

I tried to solve this question in two ways. The first, I know it is wrong. But apparently there is something wrong with the second one. It would be nice if you could help me find out what is my mistake here.
1st attempt:

k = U
1/2mv²=1/2kx²

x_f=sqrt(mv²/k)

Δx=x_f-x_i

Δx=sqrt(mv²/k)-x_i​

2nd attempt:

From conservation of energy, one can assume that the total energy of the system before the collision is equal to the total energy of the system after the collision.

E_before=E_after​

Before the collision, the mechanical energy of the target is zero. Meaning that, initially, the total energy of the system is given by the kinetic energy of the projectile, K_projectile.​

Considering that all projectile's energy was transferred to the target one may write the following equation:

K_projectile=K_target - U

1/2mv²=1/2Mv²-1/2kx²

1/2kx² =1/2Mv²-1/2mv²

x=sqrt[(Mv²-mv²)/k]
​

Thank you for your help

 

[Nathanael]

Considering that all projectile's energy was transferred to the target one may write the following equation:

K_projectile=K_target - U​

This equation implies that final kinetic energy projectile is zero, right? Your goal is to find when the compression is maximum. Is there any reason you think this would be when the projectile has no speed?

1/2mv²=1/2Mv²-1/2kx²​

You have to be careful with your equations: You used the same "v" to represent the projectile's initial velocity and the target's final velocity... But they are not the same...

 

[rude man]

Hint: invoke conservation of momentum AND conservation of energy.

 

[Ellispson]

If you're still in doubt,consider a frame of reference attached to the spring,and solve it..