Momentum and Spring Compression

[Zaft3]

I am asked to rate a series of elastic collisions in order 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 am working on the presumption 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, so 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 car's is going to relate to the time of maximum spring compression.

Any hints?

 

[Infinitum]

Hi Zaft3! Welcome to PF

I think you will need to use a bit of calculus. The impulsive force you need is only for the last instant when velocity becomes zero, and not through the whole compression of the spring. So, conserving energy you will get v' as,

$$(V')^2 = \frac{mv^2-kx^2}{m}$$

And, by impulse equation,

$$m\cdot dv' = k\cdot dt\cdot dx$$

Differentiate the first equation with respect to x, and using that relation, integrate the second equation.

 

[ehild]

I am asked to rate a series of elastic collisions in order 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.

From the instant the vehicle strikes the spring they move together till maximum compression, when the speed becomes zero, according to a simple harmonic motion.

Do you can find the time period of that SHM from the spring constant and the mass of the vehicle?

What fraction of the time period elapses from maximum speed to maximum compression (zero speed)?

ehild