How high does a rocket go?

A rocket is fired straight up, with no air and no engine after launch. With a steady $g$, it would rise $\dfrac{v^2}{2g}$. But $g$ falls off with height, so it rises higher, and at the escape speed it never comes back. You will make the program draw both heights against the launch speed, and find where the true height is $1.5$ times the steady $g$ answer.

The idea. Energy: $\tfrac12 mv^2 - \dfrac{GMm}{R} = -\dfrac{GMm}{R + h}$. With $v_e^2 = 2gR$, this gives $h = \dfrac{Rv^2}{v_e^2 - v^2}$.

Your tasks.

1. Work out the true height h for every launch speed in v.
2. Work out the steady $g$ height h_flat for every v.
3. Find the launch speed at which h is $1.5$ times h_flat.