Question

When you drop a pebble from height h, it reaches the ground with kinetic energy k if there is no air resistance. from what height should you drop it so it will reach the ground with twice as much kinetic energy?

Answer 1

-- The kinetic energy it has when it hits the ground is exactly
the potential energy it had just before you dropped it.

-- So if you want it to have twice as much kinetic energy at the bottom,
you want it to have twice as much potential energy at the top.

-- Potential energy = (mass)·(gravity)·(height).
The only one of those things that you can change is the height.

Looking at that equation, you can see that if you change the height,
the potential energy changes by the same factor.

So if you want the pebble to have twice as much potential energy,
you have to drop it from twice the original height.  (2h)