Calculating the Velocity and Distance of Two Colliding Planets using Gravity

[Hashiramasenju]

Homework Statement

Two spherical pieces of rock, of masses m1=1.0×1010kg and m2=2.0×1010kg and both of radius r=1500m (2.s.f.) are in deep space a distance of R=1000km (2.s.f.) apart. The only force between them is gravity and they are initially stationary. Find the speed of m1 at collision ? and how much distance m1 travels before colliding.

Homework Equations

F=Gm1m2/R^2
U=Gm1m2/r
E=0.5mv^2

The Attempt at a Solution

For the first part i tried to find the difference in potential then equate that to the kinetic energy but its was in vain

 

[Doc Al]

For the first part i tried to find the difference in potential then equate that to the kinetic energy but its was in vain

That's certainly part of the solution. Hint: Besides energy, what else is conserved?

Show what you did.

 

[Hashiramasenju]

That's certainly part of the solution. Hint: Besides energy, what else is conserved?

Show what you did.

momentum is conserved so m1v1=-m2v2

But how do you calculate the potential energy?

Thanks for the reply

 

[Doc Al]

momentum is conserved so m1v1=-m2v2

Right.

But how do you calculate the potential energy?

You gave the formula yourself:

U=Gm1m2/r

It's missing a minus sign, by the way.

 

[Hashiramasenju]

Right.You gave the formula yourself:

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It's missing a minus sign, by the way.

so is it Gm1m2/(2r+R)=0.5m1v1^2

where R is the distance between the surface of the two rocks

 

[Doc Al]

so is it Gm1m2/(2r+R)=0.5m1v1^2

where R is the distance between the surface of the two rocks

You need the change in potential energy, as the rocks move from initial to final position. And that will equal the total kinetic energy of both rocks, not just m1.

You need to combine that with conservation of momentum to solve for the final speed of m1.

 

[Hashiramasenju]

You need the change in potential energy, as the rocks move from initial to final position. And that will equal the total kinetic energy of both rocks, not just m1.

You need to combine that with conservation of momentum to solve for the final speed of m1.

Thats what is confusing me so is it
Gm1m2/(2r+R) -Gm1m2/(2r)=0..5m1v1^2+0.5m2v2^2

 

[Doc Al]

Thats what is confusing me so is it
Gm1m2/(2r+R) -Gm1m2/(2r)=0..5m1v1^2+0.5m2v2^2

Almost. You have the sign wrong on the left-hand side. (What you have now is negative.)

 

[jbriggs444]

Thats what is confusing me so is it
Gm1m2/(2r+R) -Gm1m2/(2r)=0..5m1v1^2+0.5m2v2^2

Since we are not told whether the separation is center to center or edge to edge, it is not clear whether you need to account for the 2r in the starting separation. In any case, to two significant figures, 2r is negligible compared to R. So you may as well simplify the equation now and save yourself some work later.

 

[Hashiramasenju]

Almost. You have the sign wrong on the left-hand side. (What you have now is negative.)

OMG ! I got the answer. Thanks alot.

So for the second part i got the answer by guessing that m1/m2=d2/d1
where d1 is the distance travveled by m1 and likewise for d2

and the answer was correct but i don't know why

 

[Doc Al]

OMG ! I got the answer. Thanks alot.

Excellent.

So for the second part i got the answer by guessing that m1/m2=d2/d1
where d1 is the distance travveled by m1 and likewise for d2

and the answer was correct but i don't know why

No guessing allowed! :-)

Hint: Where is the center of mass of this system? Does it change when they approach?

 

[Hashiramasenju]

Excellent.No guessing allowed! :-)

Hint: Where is the center of mass of this system? Does it change when they approach?

I got it. But why doesn't the CM change ?

 

[jbriggs444]

I got it. But why doesn't the CM change ?

If the two masses start at rest, the total momentum of the system is zero, yes? The total momentum of a composite system can (at least classically) also be computed as total mass multiplied by velocity of the center of mass.

If total mass is non-zero and momentum is zero, what does that tell you about the velocity of the center of mass?

 

[Hashiramasenju]

If the two masses start at rest, the total momentum of the system is zero, yes? The total momentum of a composite system can (at least classically) also be computed as total mass multiplied by velocity of the center of mass.

If total mass is non-zero and momentum is zero, what does that tell you about the velocity of the center of mass?

Thanks !