Gravitational Force?

Question

As a rocket carrying a space probe accelerates away from Earth, the fuel is being used up and the rocket's mass becomes less. When the mass of a rocket (and its fuel) is M and the distance of the rocket from the Earth's center is 1.5rE, the force of gravitational attraction between Earth and the rocket is F1.

When some fuel is consumed causing the mass to become 0.5M and the distance from Earth's center is 2.5rE, the new gravitational attraction is f2. Determine the ratio of F2 to F1.
The symbol rE is earth's radius.

Answer 1

Use the formula F = G (M1 M2) / rE^2
Substitute what you know at the two points into the equation to get two separate equations.

F1 = G*Me*M/ (1.5rE)^2
F2 = G*Me*0.5M / (2.4rE)^2
(I used Me for the mass of the earth)

The ratio of F2 to F1 will be F2 divided by F1. When you divide, all the unknown variables cancel.

Alternatively, since the force is directly proportional to the mass, halving the mass halves the force. But the force is also inversely proportional to the square of the distance. So if you move 2.5/1.5 times father away, it reduces the force by a factor of (2.5/1.5)^2. So the new force will be 0.5 * (2.5/1.5)^2 times the old force. If you do it the first way I told you, you'll see that this is exactly what you get after the unknown variables cancel.

Answer 2

Simply inverse D squared