Say I've got a rocket payload (there's no thrust, it's just dead weight) falling to earth from an altitude of 260km and I want to calculate the time it takes it to fall to 160km. It's far enough away from earth that I can't make the assumption that gravity is constant (it varies from 9.3 m s^-2 to 9.0 m s^-2). I am making the assumption that there is no drag.
I've been trying to solve the following:
G*Me / ( r(t)^2 ) = r''(t)
where G is the gravitational constant, Me is the mass of earth, r(t) is the distance from the payload to earth, and r''(t) is the acceleration.
The Attempt at a Solution
I've been trying to use laplace transforms to solve the equation but I'm a bit rusty.
I moved the r(t) ^2 over:
G*Me = r''(t) * r(t)^2
then I took the laplace transform (I'm pretty sure this is where I messed up)
G*Me/s = (s^2 R(S) - s r(0)) *R(s)^2
then I did some more algebra:
G*Me = s^3 R(s) ^3 - s^2 r(0)
and I'm not really sure where to go from here. Please help!!