at what height above the earths surface will a rocket have half the force of gravitation on it that it would have at sea level

Question

express answer in earth radii

Answer 1

Let R be the Earth's radius and M be its mass.  The gravitational force on a rocket of mass m at the surface is:

 

F = GMm/R²

 

where G is the Gravitational Constant.  The acceleration is F/m = GM/R².  Let the rocket be at a height h above the surface.  Its distance from the Earth's Center (of mass) is R + h.  We want:

 

GM/(R+h)² = (1/2)GM/R²

 

Solving for h:

 

2R² = (R+h)²

(√2)R = R + h

(√2)R - R = h

(√2 - 1)R = h

(0.414)R ≅ h

 

Note that the result is independent of the rocket's mass, m.

Answer 2

Answer deleted by Arturo. Philip already posted a good solution.