# Free fall acceleration problem

1. Apr 18, 2008

### bcjochim07

1. The problem statement, all variables and given/known data
A starship is circling a distant planet of radius R. The astronauts find that the free fall acceleration of their altitude is is half the value at the surface of the planet. How far above the surface are they orbiting? The answer should be a multiple of R.

2. Relevant equations
F=GMm/r^2 = ma

a=g=GM/r^2

3. The attempt at a solution
h is the altitude of orbit

GM/(R+h)^2 = .5* GM/R^2

2GM/(R+h)^2 = GM/R^2

2GM=(GM/R^2)(R+h)^2
2= (R+h)^2 / R^2
2R^2 = (R+h)^2
sqrt 2 * R = R+h
h= R(sqrt2 -1)

2. Apr 18, 2008

### Chi Meson

Bang on!

That is, your are right.

3. Apr 18, 2008

### tiny-tim

Hi bcjochim07!

Yes that's right!

(Why were you worried about it? )

But a bit long-winded … you could have cut it down to:

GM/(R+h)² = .5* GM/R²

so 2= (R+h)² / R²

so √2 = (R+h)/R = 1 + h/R

so h= R(√2 - 1).

Alternatively, start by saying let r be the height above the planet's centre.

Then GM/r² = .5* GM/R²,

so r = R√2, so h= R(√2 - 1).