A ball of mass m is thrown straight up from the ground level, and eventually returns to the same height. It has an initial velocity v0y. Neglect all dissipative forces. Write expressions in terms of these two quantities, and any constants you need, for the following. (NOTE: v0y should be entered in with the number zero and y (0y) as the subscript, and any answers of zero should be entered as 0.)

a. The largest kinetic energy?
b. The smallest kinetic energy?
c. The maximum potential energy, taking the initial height as the reference level.
d. The maximum height?
e. The mechanical energy.
f. The amount of work that had to be done on the ball as it was thrown.

Glad to help you with this, but only if you first try to answer these questions yourself. To get started, there is a formula for an object's kinetic energy depending on its mass and velocity - do you know what that formula is?

Kinetic energy is 1/2mv^2

Very good. So you can now answer parts a and b yourself. Now for part c - do you know the formula for gravitational potential energy?

Mgh but I don't think that's going to be the answer

Since mechanical energy is constant, and since h=0 at the start, then when all the initial KE is converted to PE (and the ball is momentarily motionless at its highest point) PE is maxed out and is equal to the initial amount of KE. So that's part (c). Part (d) you can now determine from the formula you mentioned: PE=mgh; from (c) you know PE at the max height, so you can calculate h.