A satellite fired off a planet's surface

[Joules6626]

Homework Statement

A nonrotating spherical planet with mass M and no atmosphere has radius R. A particle is fired off the surface at 3/4 the escape speed. Calculate the farthest distance it reaches from the center of the planet if it's fired of tangentially.

Homework Equations

l = r x p
U = MmG/r
F = -MmG/r^2
dKE = Fdr

The Attempt at a Solution

E1 = E2
1/2mv^2 - GMm/r = 0
1/2mv^2 = GMm/r
V(escape) = sqrt(2GM/r)
v_o = 3/4sqrt(G2M/R)

So I solved the initial velocity, but I'm confused as to what to do next. I tried this:
d1/2m(3/4sqrt(G2M/R))^2 = -MmG/r^2dr
and solving for r
so
9/32(G2M/R) = 2GM/r^3
9/32(1/R) = 1/r^3
r = cubedroot(1/(9/32*1/R)

Am I right?

 

[Joules6626]

I'm sorry, the integral should be 18/(32R) = 1/r
r = 32R/18

 

[Joules6626]

Can someone please help me with this?