Explanation:
We start by using the conservation law of energy:
[tex]\Delta{K} + \Delta{U} = 0[/tex]
or
[tex]\dfrac{1}{2}mv^2 - G\dfrac{mM}{r} = 0[/tex]
Simplifying the above equation, we get
[tex]v^2 = 2G\dfrac{M}{r}[/tex]
We can rewrite this as
[tex]v^2 = 2\left(G\dfrac{M}{r^2}\right)r[/tex]
Note that the expression inside the parenthesis is simply the acceleration due to gravity [tex]g[/tex] so we can write
[tex] v^2 = 2gr[/tex]
where [tex]v[/tex] is the launch velocity.
