# Spring compression and Momentum

I am asked to rate a series of elastic collisions in order of greatest time of max compression to least time of max compression for several vehicles with varying masses and velocities, which strike a spring with a spring constant k.

I can determine the Momentum of each case, as I am given the masses and velocities. Additionally, I can determine each of their kinetic energy.

I know that the kinetic energy of the car will be converted into potential energy in the spring:

$$1/2mv^2 =1/2kx^2$$

Also, I know the impulse of the car's is going to be

$$Ft= Δvm$$ $$t=Δvm/F$$

I also know that the Force on spring will be $F=kx$, but I am not sure how the magnitude of the momentum of the cars' is going to relate to the time of maximum spring compression. I am asking for a little guidance in my reasoning.

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$$d = A sin(\alpha t)$$
where $A$ and $\alpha$ are some constants that you need to calculate. The problem simplifies a lot if you think about the relation between the period of a harmonic oscillator and the amplitude of oscillation.