What is the ratio of the speeds v1/v2?

Question

Two satellites of masses m1 and m2 orbit a planet of mass M in circular orbits. The satellites travel in opposite directions with speeds v1 and v2. Their orbital radii are R1 and R2, respectively. Assume M>>m2>m1.

I know the answer is = square root(R2/R1)

But I don't know why.

Answer 1

This is tricky but with this problem you want to guess what kind of problem it's going to be to know where to start. We always want to start by writing down what we are given (even if there are no numbers).

We have

m₁ m₂ M R₁ R₂ v₁ and v₂

Now the key to what they gave in the problem is Circular Orbits which means that we can use the centripetal force equation. let's do that for satellite 1

F_c = m₁ v₁²/R₁

The centripetal force is equal to the net force on a moving satellite which is just the gravitational force

F_g = G M m₁/R₁²

Now let's set them equal

F_c = F_g

m₁ v₁²/R₁ = G M m₁/R₁²

Now let's pick out things we can cancel on either side, there's m₁ and R₁ on both sides so let's cancel these to get:

v₁²= G M /R₁

Now let's rearrange this equation to get just G M on one side:

v₁² R₁ = G M

Now we can do the same exact steps with satellite 2 and we get:

v₂² R₂ = G M

Now these are both equal to G M so lets just set the left sides of our two equations equal to one another:

v₁² R₁ = v₂² R₂

Next we rearrange this to get

v₁²/v₂² = R₂/R₁

Now we can square root both sides to get our solution:

v₁/v₂ = SQRT(R₂/R₁)

This is our solution, the part about opposite directions was just there to throw us off and make us think it's a momentum problem when in fact it is just a centripetal force problem