How Is the Spring Constant Calculated from Velocity and Compression Data?

[M1ZeN]

Homework Statement

The velocity of a 6 kg particle is found to be well fit by

v(t) = 15t^2 + 7 (velocity in units m/s)

a) If the particle hits a spring at t = 8 seconds, and compresses the spring 8 cm, what is the spring constant?

b) Suppose a particle with twice the mass, but the same velocity hits the (relaxed) spring. How far will the spring be compressed?

Homework Equations

Fsp = -k(s)

The Attempt at a Solution

The main problem I have with understanding this problem is one, the problem is asking for "k". I got that. It gives me "s" (8cm). I got that. I know that I have to use velocity with the time value given. The part that I'm not getting is relating the velocity equation given to finding the k unknown.

All my best.

 

[Jack21222]

You have more equations you can use.

How much energy is in the system when the particle is moving, and how much energy is in the system when the spring is compressed? How are they related?

 

[kerimek]

As a scientist, my response would be:

It seems like there may be some confusion about how to approach this problem. The given velocity equation, v(t) = 15t^2 + 7, can be used to find the acceleration of the particle at any given time using the equation a(t) = dv(t)/dt. From there, we can use the equation F = ma to find the force exerted on the spring by the particle at t = 8 seconds. This force, along with the given compression distance of 8 cm, can then be used to calculate the spring constant using the equation F = -k(s). As for part b, we can use the same process but with the new mass of the particle to find the new compression distance of the spring. I hope this helps clarify the approach to solving this problem.