What are the speeds of two Jupiter sized planets when they collide?

[AbigailG]

Homework Statement

Two Jupiter sized planets are released from rest 1.0 X 10^11 m apart. What are their speeds as they crash together?

I think my problem lies in figuring out which radius to use. In an equation like this are the radii of the planets included in the distance between them given? If not, should it be?

Homework Equations

[/B]
Ug = -Gm1m2/r

r (Jupiter) = 6.99 x 10^7 m

m (Jupiter) = 1.9 x 10^27 kg

r (between) = 1.0 X 10^11 m

The Attempt at a Solution

[/B]
Using conservation of energy:

Ei = Ef

Question #1 -- Ei is entirely Ugi...is it 2(Ug) ?

Ef = K + Ugf

Question #2 -- Is there still gravitational potential here? And if so is the radius used equal to twice the radius of Jupiter?

Question #3 -- Is the kinetic energy found for both of the planets or just one?

I need clarification on quite a few points, I apologize...but i feel like once I understand this fully I will be able to work it out myself.

 

[gneill]

For spherical objects, the distance you want to use for gravitational calculations is the distance between their centers.

Make a drawing showing the initial and final configurations. Indicate the separation between the planet centers for each situation.

The system as a whole (the system being both planets) has gravitational potential energy. You are looking for the change in potential energy between the starting configuration and the ending configuration. That change in potential energy will show up as kinetic energy for the whole system. You need to decide how that energy will be split between the components of the system.

 

[Doc Al]

In an equation like this are the radii of the planets included in the distance between them given? If not, should it be?

Compare the radii of the planets with their initial distance apart. Does it matter? (The distance is usually between their centers.)

Question #1 -- Ei is entirely Ugi...is it 2(Ug) ?

The potential energy is of the two-planet system; the formula quoted is all of it.

Question #2 -- Is there still gravitational potential here? And if so is the radius used equal to twice the radius of Jupiter?

The distance between them when they crash will equal twice the radius.

Question #3 -- Is the kinetic energy found for both of the planets or just one?

Again, the energy is of the system -- so it is the total KE of both.

(gneill beat me to it!)

 

[AbigailG]

For spherical objects, the distance you want to use for gravitational calculations is the distance between their centers.

Make a drawing showing the initial and final configurations. Indicate the separation between the planet centers for each situation.

The system as a whole (the system being both planets) has gravitational potential energy. You are looking for the change in potential energy between the starting configuration and the ending configuration. That change in potential energy will show up as kinetic energy for the whole system. You need to decide how that energy will be split between the components of the system.

Okay, so I have:

Kf - Ugf = Ugi Where: Ugi = -Gm1m2/r(between) and Ugf = -Gm1m2/2r(Jupiter)

1/2(m1+m2)-Ugf = Ugi

v = sqrt((2(ugi+ugf)/(m1+m2))
= 3 x 10^4 m/s

This is the correct answer. But why did I not have to divide the velocity amongst the two masses? Is it because I divided by the sum of the masses?

 

[Doc Al]

Kf - Ugf = Ugi

That should be a + sign.

1/2(m1+m2)-Ugf = Ugi

You left out the v^2 in your KE term. Note that your first term is the total KE of both planets!

This is the correct answer. But why did I not have to divide the velocity amongst the two masses? Is it because I divided by the sum of the masses?

You already included that "sharing" when you wrote the total KE earlier.

 

[AbigailG]

That should be a + sign.You left out the v^2 in your KE term. Note that your first term is the total KE of both planets!You already included that "sharing" when you wrote the total KE earlier.

Oops, just a typo. Sorry.

Thank you so much! I appreciate your help.