Elastic Head-On Collision, Spring Compression.

[Noxate]

Homework Statement

A 0.60kg cart moving at 5.0m/s[W] collides with a 0.80kg cart moving at 2.0m/s[E]. The collision is cushioned by a spring(k=1200N/m).

Homework Equations

[W] is Positive
P_To=P_Tf
E_To=E_Tf

The Attempt at a Solution

P_To=P_Tf
m₁v_1o+m₂v_2o=(m₁+m₂)v_f
(0.6kg)(5m/s)+(0.8kg)(-2m/s)=(0.6kg+0.8kg)V_f
3kgm/s-1.6kgm/s=(1.4kg)V_f
(1.4kgm/s)/(1.4kg)=V_f
V_f=1m/s[W]

E_To=E_Tf
1/2m₁v_1o²+1/2m₁v_2o²=1/2(m₁+m₂)v_f²+1/2kx²
1/2(0.6kg)(5m/s)+1/2(0.8kg)(-2m/s)=1/2(0.6kg+0.8kg)(1m/s²)+1/2(1200N/m)x²
7.5kgm/s+1.6kgm/s=0.24kgm/s+600N/mx²
9.1kgm/s-0.24kgm/s=600Nmx²
0.0148=x²
x=0.12m

Im pretty sure this is correct. I think I actually figured out what my real question about this was while typing this out. Initially when I saw this problem I wanted to just take the combined initial energy of both carts 9.1J and use that equal to elastic potential energy, so 1/2kx²=9.1J. I assumed that since the max compression should happen in the moment before both carts change direction or the spring rebounds that the velocity of both carts should be 0. Now I think the kinetic energy in the E_Tf is actually the kinetic energy of the entire "system" of both carts and the spring moving since the cart moving West would push everything slightly west. And so the carts do have a velocity of 0 relative to the spring at max compression but the entire collection of stuff has a velocity relative to the Earth or ground. Is that somewhat right? The only other thing that bothers me about this is why would the kinetic energy of all of them only be 1/2(m₂+m₂)V_f² shouldn't this include the mass of the spring if the spring was also moving?

 

[haruspex]

the max compression should happen in the moment before both carts change direction or the spring rebounds that the velocity of both carts should be 0.

No, you can't assume that.

the carts do have a velocity of 0 relative to the spring at max compression but the entire collection of stuff has a velocity relative to the Earth or ground.

That's right. The max compression is when the rate of change of spring length is zero, and that occurs when the carts are moving at the same velocity.

The only other thing that bothers me about this is why would the kinetic energy of all of them only be 1/2(m₂+m₂)V_f² shouldn't this include the mass of the spring if the spring was also moving?

It would, but you are not given a mass for the spring, nor how the spring was moving beforehand, so you have no choice but to assume that either it is massless or it is included in the mass of one of the carts.