Question

A 2500-kg rocket blasts off from Planet A headed directly towards Planet B. The masses of Planet A and B are 6x1024 kg and 3x1025 kg respectively and the distance between the planets is 4x108 m. How far from Planet A will the rocket have traveled when the net gravitational force on the rocket from the two planets is zero?

Answer 1

Planet A has a mass of 6x10^24 kg.
Planet B has a mass of 3 x 10^25kg.
Rocket has a mass of 2500kg.

Force on rocket from planet A = G(6x10^24kg)(2500kg)/r^2 where r is the distance from planet A.

Force on rocket from planet B = G(3 x 10^25kg)(2500kg)/R^2 where R is the distance from planet B.

The net force is zero then when:
G(3 x 10^25kg)(2500kg)/R^2 = G(6x10^24kg)(2500kg)/r^2

clearly we can cancel G and the mass of the ship from both sides;

(3 x 10^25kg)/R^2 = (6x10^24kg)/r^2

Which gives us one equation in two variables, however we have one more piece of r+R = 4x10^8 m

We now have two equations and two variables. Let's solve for one of them:

r = 4x10^8 - R and substitute back:

(3 x 10^25kg)/R^2 = (6x10^24kg)/(4x10^8 - R)^2

Multiplying by R^2 and dividing by (6x10^24kg)
(3 x 10^25kg)/(6x10^24kg) =R^2 / (4x10^8 - R)^2

5 =R^2 / (4x10^8 - R)^2
?5 = R/(4x10^8 - R)
(4x10^8 - R)?5 = R

(4x10^8)?5 - R?5 = R
(4x10^8)?5= R + R?5
(4x10^8)?5 = R(1+?5)
(4x10^8)?5 / (1+?5) = R

R = 2.76 x 10^8 m from planet B.
r = 4x10^8 - 2.76 x 10^8

r=1.24 x 10^8 m from planet A. This is 124 million meters or 124000 km.

There was an easier way to do this but I wanted you to see it done from first principles.