How to Calculate the Kinetic Energy of a Moving Spring?

[makeAwish]

Homework Statement

We usually ignore the kinetic energy of the moving coils of a spring, but let's try to get a reasonable approximation to this. Consider a spring of mass M, equilibrium length L0, and spring constant k. The work done to stretch or compress the spring by a distance L is 0.5kx^2, where x = L – L0.
(a) Consider a spring, as described above that has one end fixed and the other end moving with speed v. Assume that the speed of points along the length of the spring varies linearly with distance l from the fixed end. Assume also that the mass M of the spring is distributed uniformly along the length of the spring. Calculate the kinetic energy of the spring in terms of M and v.
(Hint: Divide the spring into pieces of length dl; find the speed of each piece in terms of l, v, and L; find the mass of each piece in terms of dl, M, and L; and integrate from 0 to L. The result is not 0.5Mv^2, since not all of the spring moves with the same speed.)


The attempt at a solution

v = (qL^2)/2 where q is the constant proportionality of v and l
m = (M^2)/(2 landa) where landa is the linear mass density

I'm not sure if my current workings are correct. And how to get rid of these constants?

 

[lanedance]

this in an integral question

you need to show you got your equations, try writing down an equation for the energy of an infinetesimal piece of the spring ie. dE in terms of v(x) and dm, then think how dm is related to dx