# 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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## 1 Answer

The question seems a bit odd because "time of maximum spring compression" is an odd concept. The spring compression is a function of time and the time of maximum spring compression is zero because it's an instant not a time interval. Maybe the question means the time interval from the time the car first touches the spring to the time of greatest compression.

Assuming this is the case, and bearing in mind that because this is a homework question we're only allowed to give hints, the trick to doing this question is to realise that the spring behaves as a simple harmonic oscillator i.e. the compression of the spring from the moment the car touches it will be:

$$d = A sin(\alpha t)$$

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.

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I see now that the question is really asking me if the time to maximum spring compression varies depending on the momentum of each vehicle, or if it stays the same regardless. As I am not at yet at a level to grasp the equation stated, I reasoned that the vehicle with the greatest momentum would result in the greatest time interval to maximum spring compression. Admittedly, this decision was little more then guessing, but unfortunately the assignment was due and I had to make a decision. That being said, I would still like to understand the truth behind this question. –  Kurt Jun 12 '12 at 16:55
It's not the vehicle momentum, it's just the mass. Once the vehicle hits the spring you have a spring + mass oscillator, and the period is sqrt(M/k) where k is the spring constant and M is the mass. k is constant so the period, and therefore the time to maximum compression is proportional to sqrt(M). See en.wikipedia.org/wiki/… –  John Rennie Jun 12 '12 at 17:09