Question

If i run down some stairs and stop, what happens to your kinetic energy and your initial gravitational potential energy?

Answer 1

First, look at the two types of energy you're asking about:

Kinetic Energy depends on the mass and speed of the object
`KE=1/2mv^2` while Gravitational Potential Energy depends on the mass and position of the object `U_g` `= mgh`.

So in your example, by coming to a stop your kinetic energy becomes zero. Because your height decreased, your potential energy also decreases (whether it equals 0 or not depends on where you define 0 `U_g` to be.)

All of the energy you "lost" has been converted into internal energy, sound energy and heat.