How the fuck does one...

A 0.29 kg ladle sliding on a horizontal frictionless surface is attached to one end of a horizontal spring (k = 435 N/m) whose other end is fixed. The ladle has a kinetic energy of 10 J as it passes through its equilibrium position (the point at which the spring force is zero).
At what rate is the spring doing work on the ladle when the spring is compressed 0.10 m and the ladle is moving away from the equilibrium position? (in Watts)

W = int(Fdr)
dw/dt = F dr/dt = Fv  (The rate of work is the spring force times the velocity)

dw/dt = mkv  (You know m and k, now you just need v)

W = int(Fdr) = int(kx dx) = 1/2 k x^2
That integral is going from -.1 to 0
W = 1/2 k (.1)^2 = k/200 = change in KE = KEf - KEi
k/200 = KEf - KEi

KEi = 10 - k/200 (Because KEf = 10j)

1/2 m V^2 = 10 - k/200
V^2 = 2/m(10-k/200)
v = sqrt[2/m(10-k/200]

dw/dt = m*k*sqrt[2/m(10-k/200]