Orbit radius of 2 collided satellites

[adiggy93]

Homework Statement

Two satellites have mistakenly been put in identical Earth orbits of radius R. Satellite A with mass "m" is orbiting clockwise while satellite B with mass "2m" is orbiting counterclocwise. The two satellites have a head on collision and move as one body after the collision. In terms on the given quantities, what orbit radius would the collided objects have to possesses in order to move in a circular path?


2. Homework Equations /Attempt
I believe that since the satellites are orbiting at the same radius, the are orbiting at the same speed so
Va = - Vb

I can get a quarter of it but I'm not even sure if it's right:

(3m)(vf)²/(Rf) = ?

using the collision formula to get vf
mVa + 2mVb = 3m Vf
Vf = (Va - 2Vb)/3

so (3m)[(Va - Vb)/3]/Rf = ?

I'm very stuck. Help!

 

[benhou]

Ok, do you know these formulas?
$$K_{s}=\frac{mv^{2}}{2}= \frac{GMm}{2r}$$

 

[adiggy93]

The formulas that I learned were mv^2/r = GMm/r^2

 

[benhou]

Well, they are the same thing. Since you think the satellites before they collide have the same speed, use your formula to check it.

 

[adiggy93]

but how? the masses of the satellites cancel out and the radius of orbit is unkown.

 

[benhou]

but how? the masses of the satellites cancel out and the radius of orbit is unkown.

Exactly, the masses cancel out. The speed is not dependent on the mass. Therefore, their speeds are the same.

 

[adiggy93]

Ah ok. But now I have to find the orbit radius but in terms of the quantities given to me in the problem, which were just m & 2m.

 

[benhou]

Before you go on with the orbit radius, why don't you plug in the speed since you know they are the same?

 

[adiggy93]

By plugging the speed in do you just mean putting in the variable and manipulating the formula so the radius is isolated? Because the speed itself is unknown even though I know the speeds are the same.

 

[benhou]

Sorry, should have been more clear. According to your formula, if we want to find r, we have to first find V. Now we know $$v_{a}=v_{b}$$, go back to the Conservation of Momentum formula or the collision formula and find V.

 

[adiggy93]

a friend of mine came up with this as the answer. I'm not sure if it's right though.

Va = - Vb
so let's just write Vi as V

V²/R = GM/R²
V = √(GM/R)

now using the formula for an inelastic equation to find Vf

mV - 2mV = 3mVf
Vf = -V/3

replace v with √(GM/Ri)
Vf = -[√(GM/Ri)]/3

Then move on the the new mass, AB, the combined satellites, with a mass of 3m:

(3m)Vf²/Rf = (3m) GM/Rf²
Vf²/Rf = GM/Rf²

replace Vf with what you found above

[-(√(GM/Ri))/3]²/Rf = GM/Rf²

simplify

[(GM/Ri)/9]/Rf = GM/Rf²

simplify more

(GM/9Ri)/Rf = GM/Rf²

and more

(GM)(Rf²) = (GM)(9Ri)(Rf)

cancel out the Gmp's and then divide each side by Rf and so you get to

Rf = 9Ri

 

[benhou]

Looks good for me.

 

[adiggy93]

alright. thank you so much for the help :)