Time derivative of gravity due to acceleration

[Ofinns]

Homework Statement

We have the equation for gravity due to the acceleration a = -GM/r², calculate velocity and position dependent on time and show that v/x = √2GM/r_0³⋅(r/r₀-1)

Homework Equations

x(t = 0) = x₀ and v(t = 0) = 0

The Attempt at a Solution

v = -GM∫1/r² dt
v = dr/dt
v² = -GM∫1/r² dt⋅dr/dt = √Gm/r

for x... dt/dt = √GM/r²
r depends on t so r(t)?
break out and integrate on both sides
x = 3/2^2/3⋅(√GMt+c)^2/3 which obviously is not so good as it is time dependent.

Where do I go wrong? How should I proceed?

 

[tnich]

Homework Statement

We have the equation for gravity due to the acceleration a = -GM/r², calculate velocity and position dependent on time and show that v/x = √2GM/r_0³⋅(r/r₀-1)

Homework Equations

x(t = 0) = x₀ and v(t = 0) = 0

The Attempt at a Solution

v = -GM∫1/r² dt
v = dr/dt
v² = -GM∫1/r² dt⋅dr/dt = √Gm/r

for x... dt/dt = √GM/r²
r depends on t so r(t)?
break out and integrate on both sides
x = 3/2^2/3⋅(√GMt+c)^2/3 which obviously is not so good as it is time dependent.

Where do I go wrong? How should I proceed?

It's hard to tell where you have gone wrong when I can't tell what you have done. I'm not sure what x... dt/dt = √GM/r² means. Could add in a few more steps to show what you mean? Please use LaTeX. It is easy to use and makes it easy to express what you mean.