Deriving the Force-Position Relationship for a Spring Powered Model Car

[prodigy803]

Homework Statement

In your design of an experimental spring powered model car, you note that the speed of
the car (mass Mc) increases as the car travels further. The exact relationship is that v(x) =
C|x|, where C is a constant and x is the position of the car with respect to the starting
position.

a) Derive an expression for the force provided by the spring as a function of distance.
b) How much work does the spring do as it moves the cart from x0 to xf?

Homework Equations

F=ma
W = Fd cosθ
W spring = ΔKE = 1/2*k*(x_f² - x_i²)

The Attempt at a Solution

I thought of integrating the velocity to get the position function but since the velocity given is the velocity as a function of position, I'm not exactly sure where to go with that. Any advice would be great! Thanks!

 

[Doc Al]

How might you figure out the acceleration, given the velocity function?

 

[prodigy803]

Well acceleration is dv/dt. But since the velocity given is a function of position would it be equivalent? Setup the equation as v(X) = dv/dt?

Giving me v(X)dt = dv, then integrate with t_fand t₀ as the limits?

 

[prodigy803]

Or can I take the derivative with respect to position rather than time?

For example:
d/dx [v(X)] = d/dx [C|x|]

I'm just not confident in doing so for some reason. Just because everything is usually taken with respect to time.

 

[Doc Al]

Well acceleration is dv/dt. But since the velocity given is a function of position would it be equivalent?

You'll take the derivative (d/dt) of the velocity function.

Setup the equation as v(X) = dv/dt?

I think you mean a = dv/dt. Try it!