Question

Planet A and Planet B are in circular orbits around a distant star. Planet A is 6.0 times farther from the star than Planet B. Find the ratio of their speeds. Va/Vb.

Answer 1

The ratio of speed will be 0.408

Step-by-step explanation:

We have given that Planet A is 6 times farther than planet B

So
`R_A=6R_B`

We know that speed is given by
`v_A=\sqrt{(GM)/(R_A)}`, here G is gravitational constant and
`R_A` is distance from star to planet A

As
`R_A=6R_B`

So
`v_A=\sqrt{(GM)/(6R_B)}`-----EQN 1

Speed of planet B
`v_B=\sqrt{(GM)/(R_B)}`------RQN 2

Dividing equation 1 by equation 2

`(v_A)/(V_B)=\sqrt{(1)/(6)}`

`(v_A)/(V_B)=0.408`

So the ratio of speed will be 0.408

Answer 2

The answer is easy if you know the physics.



a/a' = (r/R)^2 = (r/9r)^2 = 1/81; so that a = a'/81.
A's radial acceleration must be 1/81 of B's. And each acceleration is a = v^2/R and a' = V^2/r, where v and V are the tangential speeds you want the ratio for.

a = v^2/R = v^2/9r = V^2/81r = a'/81 In which case v^2/V^2 = 9/81 = 1/9; so that 1/3 = Va/Vb ANS