Satellite A of mass m orbits a planet of mass M at a distance of R from the center of the planet. The speed of the satellite is v. Answer in terms of m, M, R and v.

Question

What is the speed of a satellite B whose mass is 2 m? The distance from the center of the planet to the satellite B is R and the mass of the planet is M. (The answer is v I just don't know how to get there)

Answer 1

Are you familiar with the vis-viva equation?

v = √[GM(2/r - 1/a)]

r = distance from satellite to center of planet

a = semi-major axis of elliptical orbit

For a circular orbit of radius R,

r = a = R

Then the vis-viva equation simplifies to

v = √(GM/R)

Note v does not depend on the mass or the satellite; only on the radius of its orbit and the mass of the planet. Therefore if 2 different satellites are orbiting the same planet in circular obits of the same radius, their orbital speeds will be the same, regardless of the masses of the satellites. That is the situation in this problem.

Answer 2

For satelite A we have: mv²/R = GmM/R² (1),

For satelite B the same equation: 2mv_B²/R = G2mM/R² (2).

Reducing (1) by m/R we get v² = GM/R (1').

Reducing (2) by 2m/R we get v_B² = GM/R (2').

From (1') and (2') we have v_B² = v² and v_B = v.

Here G is gravitational constant.

Suttelite B has the same speed as satelite A.

Answer 3

Since the only force acting on the satellite in circular orbit is the gravitational force of the planet M on the satellite m which distance from the center of the planet is R. The satellite is not falling toward the planet, but is constantly falling around the planet and in a circular orbit the acceleration is given by the formula a=v^2/R and it is directed toward the center of the planet. Now according to Newton second law of gravitation F=ma and F=GMm/R^2 are equal to each other: GMm/R^2=mv^2/R from which simplifying and solving for v gives

v=(GM/r)^.5

Now if the mass of the satellite B is 2m the speed is going to be the same! because the mass of the satellite does not appear in the formula...