What is the Maximum Height Attained by an Object Launched from Earth's Surface?

[GOsuchessplayer]

Homework Statement

An Object is fired vertically upward from the surface of the Earth ( of radius R_E ) with an initial speed v_i that is comparable to but less than the escape speed v_esc

Show that the object attains a maximum height h given by h=$$\stackrel{R_{E}v^{2}_{i}}{v^{2}_{esc}-v^{2}_{i}}$$

Homework Equations

The Attempt at a Solution

I have the solution but I am confused with a step.
Here is the solution provided up until the step on which I am confused:

$$(K+U_{g})_{f}$$=$$(K+U_{g})_{f}$$

$$\stackrel{1}{2}$$m$$v^{2}_{i}$$ - $$\stackrel{GmM_{E}}{R_{E}}$$ = 0 - $$\stackrel{GmM_{E}}{R_{E}+h}$$

where $$\stackrel{1}{2}$$m$$v^{2}_{esc}$$= $$\stackrel{GmM_{E}}{R_{E}}$$

Then$$\stackrel{1}{2}$$$$v^{2}_{i}$$ - $$\stackrel{1}{2}$$$$v^{2}_{esc}$$= -$$\stackrel{1}{2}$$$$v^{2}_{esc}$$ ($$\stackrel{R_{E}}{R_{E}+h}$$)

So the jump I am confused about is the equation that follows the "Then" Statement. I'm not sure why this is true.

(sorry for the poor look of the equations. Hopefully its readable, I wasn't sure how to make fractions.)

 

[Borek]

\frac x y

$$\frac x y$$

And if not absolutely necessary don't put LaTeX inline, it looks better in its own lines.