A satellite fired off a planet's surface

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A particle is fired tangentially from a nonrotating spherical planet at 3/4 of the escape speed, and the goal is to determine the farthest distance it reaches from the planet's center. The relevant equations include gravitational potential energy and kinetic energy, with the escape velocity derived from the planet's mass and radius. The user attempts to calculate the distance using energy conservation principles but expresses confusion regarding the integration process. They derive an equation relating the distance to the planet's radius but seek confirmation of their calculations. Clarification on the integration steps and final distance calculation is requested to ensure accuracy.
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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?
 
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I'm sorry, the integral should be 18/(32R) = 1/r
r = 32R/18
 
Can someone please help me with this?
 

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