JacksonL said:
The work energy theorem is:
1/2mv^2-1/2mv^2.
That doesn't make any sense. The work energy theorem is an equation, and should have a left hand side and an equal sign. And 1/2mv^2-1/2mv^2 = 0. Please put the relevant subscripts.
At any rate, you don't need that form of the work-energy theorem here.
You do need this, though:
Translational kinetic energy is 1/2mv^2
And this is written correctly. You don't need v_1 and v_2 here, because the ball starts at rest. Just use one variable for the speed at which the ball is launched, v.
When solving this, consider the spring and ball as a single system. You compress the spring with the ball on top of it, release it, the spring stretches to its original length carrying the ball with it, following which the ball continues to move vertically upward (it's "launched"). Now ask yourself the following:
1) When the spring is fully compressed, what form of energy resides in the system? How do you calculate this?
2) As the spring is released and is relaxing to its original length, what energy transformations are taking place? What energy is decreasing? What's (or what are) increasing?
3) Finally, when the spring attains its original length, what forms of energy exist in the system?
Equate the energy in state 1 to that in state 3 with Conservation of Energy. Now write down the relevant equations. You can now solve for the initial kinetic energy and thereby, the initial upward speed of the ball at the point of launch.
Continuing on, the ball travels upward until it reaches a maximum height, when it becomes momentarily stationary.
4) What form of energy exists in the ball at this time? Again use conservation of energy to solve for the maximum height the ball attains. Remember that this height is taken from the point of launching. The question is asking for how high the ball gets from its *original* position. What do you need to do?
