Gravitational equation of motion for an object falling on a planet.

Suppose mass of planet is M and mass of object is m.
Object is r meters away from planet which is R+H where R is radius of planet and H is height.

At any instance, object is x meters away from centre of mass of planet.

According to Sir Issac Newton
Gravitational force on object = ma = - GMm/x²
a = - GM/x²
dv/dt = - GM/x²
dv/dt dx/dx = - GM/x²
dx/dt dv = - GM/x² dx
v dv = - GM/x² dx

Integrating both sides

v²/2 = GM/x + C

At x = r, v = u

u²/2 = GM/r + C
C = u²/2 - GM/r
 = (ru² - 2GM)/2r

Now,
v² = 2(GM/x + C)
v = - sqrt(2) sqrt((GM + xC)/x)
dx/dt = - sqrt(2) sqrt((GM + xC)/x)
dt = - 1/sqrt(2) sqrt(x/(GM + xC)) dx

Let xC = GM tan²c
x = GM/C tan²c
dx = GM/C 2 tan(c) sec²c dc
  = 2GM/C tan(c) sec²c dc

tan²c = xC/GM
tan(c) = sqrt(xC/GM)
sec²c = xc/GM + 1
sec(c) = sqrt((xC + GM)/GM)

Putting in equation

dt = - sqrt(GM/2C) sqrt(tan²c/(GM + GM tan²c)) 2GM/C tan(c) sec²c dc
  = - sqrt(2) GM/(C sqrt(C)) sqrt(tan²c / sec²c) tan(c) sec²c dc
  = - sqrt(2) GM/(C sqrt(C)) tan²c sec(c) dc
  = - sqrt(2) GM/(C sqrt(C)) (sec²c - 1) sec(c) dc
  = - sqrt(2) GM/(C sqrt(C)) (sec³c - sec(c)) dc 

Integrating both sides

t = - sqrt(2) GM/(C sqrt(C)) (1/2 sec(c) tan(c) + 1/2 ln|sec(c) + tan(c)| - ln|sec(c) + tan(c)|) + C₂
  = - sqrt(2) GM/(C sqrt(C)) (1/2 sec(c) tan(c) - 1/2 ln|sec(c) + tan(c)|) + C₂
  = - GM/(C sqrt(2C)) (sec(c) tan(c) - ln|sec(c) + tan(c)|) + C₂
  = - GM/(C sqrt(2C)) (sqrt((xC + GM)/GM) sqrt(xC/GM) - ln|sqrt((xC + GM)/GM) + sqrt(xC/GM)|) + C₂

At t = 0, x = r

0 = - GM/(C sqrt(2C)) (sqrt((rC + GM)/GM) sqrt(rC/GM) - ln|sqrt((rC + GM)/GM) + sqrt(rC/GM)|) + C₂
C₂ = GM/(C sqrt(2C)) (- ln|sqrt((rC + GM)/GM) + sqrt(rC/GM)| + sqrt((rC + GM)/GM) sqrt(rC/GM))

Putting in equation

t = GM/(C sqrt(2C)) (- sqrt((xC + GM)/GM) sqrt(xC/GM) + ln|sqrt((xC + GM)/GM) + sqrt(xC/GM)| - ln|sqrt((rC + GM)/GM) + sqrt(rC/GM)| + sqrt((rC + GM)/GM) sqrt(rC/GM))

At collision, x = R

t = GM/(C sqrt(2C)) (- sqrt((RC + GM)/GM) sqrt(RC/GM) + ln|sqrt((RC + GM)/GM) + sqrt(RC/GM)| - ln|sqrt((rC + GM)/GM) + sqrt(rC/GM)| + sqrt((rC + GM)/GM) sqrt(rC/GM))

Collision velocity

v = - sqrt(2) sqrt((GM + RC)/R)
  = - sqrt(2) sqrt(((2rGM + R(r u² - 2GM)/2r)/R)
  = - sqrt(2) sqrt((2rGM + Rru² - 2RGM)/2Rr)
  = - sqrt((2(R + H)GM + R (R + H)u² - 2RGM)/R(R + H))
  = - sqrt((2RGM + 2HGM + R² u² + RHu² - 2RGM)/R(R + H))
  = - sqrt((2HGM + R²u² + RHu²)/R(R + H))

These are gravitational equation of motion for a small object.

Thank you.
Praveen Kumar Sirohiwal