Elastic Head-On Collision, Spring Compression.

In summary, 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 with a spring constant of 1200N/m. Using the equations for conservation of momentum and conservation of energy, the final velocity of the combined system is determined to be 1m/s[W] and the maximum compression of the spring is found to be 0.12m.
  • #1
Noxate
2
0

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
PTo=PTf
ETo=ETf

The Attempt at a Solution


PTo=PTf
m1v1o+m2v2o=(m1+m2)vf
(0.6kg)(5m/s)+(0.8kg)(-2m/s)=(0.6kg+0.8kg)Vf
3kgm/s-1.6kgm/s=(1.4kg)Vf
(1.4kgm/s)/(1.4kg)=Vf
Vf=1m/s[W]

ETo=ETf
1/2m1v1o2+1/2m1v2o2=1/2(m1+m2)vf2+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 ETf 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)Vf2 shouldn't this include the mass of the spring if the spring was also moving?
 
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  • #2
Noxate said:
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.
Noxate said:
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.
Noxate said:
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)Vf2 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.
 

1. What is an elastic head-on collision?

An elastic head-on collision is a type of collision between two objects where both objects are moving towards each other and collide without any deformation or loss of kinetic energy.

2. What is spring compression?

Spring compression is the displacement or reduction of size of a spring due to an external force or load applied to it. This causes the spring to store potential energy.

3. How does a spring affect an elastic head-on collision?

In an elastic head-on collision, the spring acts as a restoring force that causes the colliding objects to bounce back after the collision. This results in the conservation of kinetic energy during the collision.

4. What factors affect the amount of spring compression in an elastic head-on collision?

The amount of spring compression in an elastic head-on collision is affected by the mass of the colliding objects, the speed at which they are moving towards each other, and the stiffness of the spring.

5. How is the amount of kinetic energy conserved in an elastic head-on collision?

In an elastic head-on collision, the amount of kinetic energy before the collision is equal to the amount of kinetic energy after the collision. This is because no energy is lost during the collision due to the elastic nature of the objects and the spring.

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