Question

two satellites A and B go around the planet P in circular orbit having radius 4R and R. If the speed of the satellite A is 3v. Then find the speed of the satellite B

Answer 1

circular orbit with uniform motion.

let the satellite with the orbiting radius 4 R  have a mass m1.
   its speed in the orbit = 3 v.

let the satellite with the orbit radius R have a mass m2.
   let its speed = v2.

let the mass of planet = M

The gravitational force of attraction is equal to the centripetal force acting on the satellite making it go around in a circle.

       centripetal force formula is = m V²/r          --- (1)

satellite 1:                 m1 * (3v)² / (4 R) =  G M m1 / (4 R)²        --- (2)
                         v² = G M / (36 R)          --- (3)

satellite 2:                m2 * v2² / R  =  G M m2 / R²        --- (4)
                                 v2² = G M / R          ---(5)
 
     from (3),    GM /R = 36 v²

     from (5),    v2² = 36 v²
                             v2 = 6 v

Answer 2

F=GMm/R2,

now force of Gravitation provide centripetal force

therefore mv2/R=GMm/R2

v=[GM/R]1/2

Now, according to question

3v=[GM/4R]1/2

3v=1/2[GM/R]1/2 ----------(1)

and

let the velocity of the other planet be x,

x=[GM/R]1/2 -------------(2)

from (1) and (2)

3v=x/2,

x=6v

hence solved