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 each of the following in terms of m, M, R and v.

Question

(b) Satellite B of mass m has orbital radius R. Satellite B orbits a planet of mass 4 M. Whatis the speed of the satellite B?

(c)Satellite B of mass m orbits a planet of mass M with speed 2 v. What is the radius of thesatellite B’s orbit?

(d)How fast should satellite B of mass m orbit a planet of mass M so that its orbital radius is 8 R?

(answer for b is 2V, answer for c is 1/4 R, answer for D is V/sqrt8

i dont know how to get these answers)

Answer 1

The satellite's centripetal acceleration

v²/R

must equal its gravitational acceleration

GM/R²

The reason is that the centripetal acceleration is caused by gravity.

After setting those two equal we can derive the following

v = sqrt(GM/R) (1)

R = GM/v² (2)

(b) From equation (1), v depends on the square root of M.

Therefore multiplying M by 4 causes v to double.

(c) From equation (2) we see that R is inversely proportional to the square of v.

Therefore doubling v, causes the radius to be reduced by a factor of 4.

(d) From equation (1), v is inversely proportional to the square root of R.

Therefore multiplying R by 8 reduces v by the factor of sqrt(8).