Spring Launch

We are going to use the fact that the change in an object's kinetic energy is equal to the work done on it by external forces. Initially, the ball is at rest, so its kinetic energy is zero. In the final state, the ball has reached its maximum height so the kinetic energy is zero there as well. Thus the net work done on the ball by external forces must be zero. There are two forces acting on the object: gravity and the spring force. The spring force acts in the direction of motion, so that it speeds up the object and does work. The amount of work that it does is Ws=12k(Δy)2

. On the other hand gravity acts in the direction opposite to velocity, and so it contributes a negative work. The work due to gravity is Wg=−mg(hmax+Δy).
For the total work to be zero we must have Ws=−Wg
or 12k(Δy)2=mg(hmax+Δy). Solving for hmax, we find hmax=k2mg(Δy)2−Δy=Δy(kΔy2mg−1).