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}mv^{2}_{i} - \stackrel{GmM_{E}}{R_{E}} = 0 - \stackrel{GmM_{E}}{R_{E}+h}

where \stackrel{1}{2}mv^{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.