Solving 2-Sphere Collision Problem: M, 2M, R, & 2R

[whatisphys]

Homework Statement

Two spheres having masses M (sphere 1) and 2M (sphere 2) and radii R and 2R, respectively, are released from rest when the distance between their centers is 8R.
How fast will sphere 1 be moving when they collide? Assume that the two spheres interact only with each other. Enter your answer in units of sqrt(GM/R).
How fast will sphere 2 be moving when they collide? Enter your answer in units of sqrt(GM/R)

Homework Equations

-Gmm/r
(mv^2)/2
m1v1 + m2v2 = 0 since starts at rest

The Attempt at a Solution

Okay.
So what I did was I first calculated initial PE and Final PE. I then calculated the change in PE which turned out to be -5GM^2/12R
Then, I equated it to deltaKE = -deltaPE which i got as (Mv1^2)/2 + (2Mv2^2)/2 = 5GM/12R.
I used conservation of momentum to find out the ratio of V1 to V2 which was V2 = -1/2(V1)
After that I substituted that to the equation above and solved for V1 as 20GM/36R.
But it says it is wrong. Any help would be appreciated. Thank you

 

[Simon Bridge]

Please show your working for the change in potential energy.

 

[whatisphys]

PEi = GM2M/8R = -GM^2/4R and PEf = GM2M/3R = -2GM^2/3R

delta PE = Final - initial

make them so that they have common base which is -8GM^2/12R + 3GM^2/12R

 

[Simon Bridge]

OK I see ... so for conservaton of energy and momentum respectively you got:

$$v_1^2 + 4v_2^2 = \frac{10}{12}\frac{GM}{R}\\
v_1 + 2v_2 = 0$$ ... after dividing through by M in both equations and multiplying through by 2 in the top one.
That about right?

After that it is solved by simultaneous equations.
... your reasoning seems sound, so you need to check your algebra.

 

[HalfLostAndSearching]

.

Your approach is on the right track, but there are a few errors in your calculations. First, the initial potential energy should be calculated as -GM^2/8R, not -GM^2/12R. This is because the distance between the centers of the spheres is 8R, not 12R.

Secondly, when you equate deltaKE = -deltaPE, you should have (Mv1^2)/2 + (2Mv2^2)/2 = (GM^2/8R) - (-GM^2/12R), since you are looking at the change in energy, not just the final energy.

Lastly, when you use conservation of momentum, the equation should be m1v1 + m2v2 = 0, not just m1v1 + m2v2. This is because the initial momentum is zero, since the spheres start at rest.

Correcting these errors, you should get the following equations:
(Mv1^2)/2 + (2Mv2^2)/2 = (GM^2/8R) - (-GM^2/12R)
m1v1 + m2v2 = 0

Solving these equations, you should get v1 = sqrt(30GM/36R) and v2 = -sqrt(15GM/36R). However, since the negative velocity doesn't make sense physically, we can discard it and get the final answer for v1 as sqrt(30GM/36R) = sqrt(5GM/6R) and v2 as sqrt(15GM/36R) = sqrt(5GM/12R).

These answers are in the correct units of sqrt(GM/R), so you should be good to go. Keep in mind that when you are working with conservation of energy and momentum, you need to be careful with the signs and make sure you are taking into account the initial and final states of the system. Hope this helps!