How Do You Calculate Maximum Spring Compression in an Inelastic Collision?

[darklich21]

Homework Statement

A 10500kg freight car rests against a spring bumper at the end of a raildroad track. The spring has constant k = 3.4 x 10^N/m. The car is hit by a second car of 9400 kg mass moving at 7.5 m/s, and the two cars couple together.

a) what is the maximum compression of the spring?
b) what is the speed of the two cars together when the rebound from the spring?

Homework Equations

The Attempt at a Solution

For starters, I know this is an elastic collision problem since the 2 cars stick together. Perhaps the equation m1v1=V(m1+m2) has some relevance. I also know the initial energy of the first car is 0 since it's at rest. But I'm not sure how to factor the spring here. What equation or equations can I use to solve this problem?

 

[darklich21]

Ok correction, I solved part b. I did use m1v1+ m2v2=v(m1+m2), and I got 3.5 m/s, which is correct. But how do I solve part a? What equation should I use

 

[tomcue]

As a scientist, it is important to approach problems systematically and use relevant equations to solve them. In this case, we can use the conservation of momentum and energy principles to solve for the maximum compression of the spring and the speed of the two cars after the rebound.

First, we can use the conservation of momentum principle, which states that the total momentum of a closed system remains constant. In this case, the initial momentum of the second car, 9400 kg x 7.5 m/s = 70500 kgm/s, will be equal to the final momentum of the two cars together, which we can represent as (10500 kg + 9400 kg) x V, where V is the velocity of the two cars together. Solving for V, we get V = 4.5 m/s.

Next, we can use the conservation of energy principle, which states that the total energy of a closed system remains constant. In this case, the initial energy of the system is solely in the form of kinetic energy from the second car, which we can represent as 1/2 x 9400 kg x (7.5 m/s)^2 = 264375 J. The final energy of the system will be in the form of potential energy from the compressed spring, which we can represent as 1/2 x k x x^2, where k is the spring constant and x is the maximum compression of the spring. Setting these two equal, we can solve for x and get x = 2.19 m.

Therefore, the maximum compression of the spring is 2.19 m and the speed of the two cars together after the rebound is 4.5 m/s.