Two Insulating Spheres in Each Other's Electric Field

faceoclock

Hi, I'd like to ask the good people of this forum for some help.
Here's a problem I've been working on for a while, and I'm seriously at my wit's end. I guess there's something I'm missing here...


1. The problem statement, all variables and given/known data
Two insulating spheres have radii r₁ and r₂, masses m₁ and m₂, and uniformly distributed charges -q₁ and q₂. They are released from rest when their centers are separated by a distance d. How fast is each moving when they collide? Suggestion: Consider conservation of energy and of linear momentum.

2. Relevant equations
I thought these were relevant:
Momentum=mv
Kinetic energy = 1/2(mv^2)
[tex]\Delta[/tex]U = -q[tex]\int[/tex]E dr

3. The attempt at a solution
First I solved for the potential energy that this system gains when the two spheres are moved apart:
[tex]\Delta[/tex]U = q1[tex]\int^{d}_{d-r1-r2}[/tex]E dr = k(q1)(q2)([tex]\frac{1}{d-r1-r2}[/tex] - 1/d)

I figured this is the amount of energy the spheres would have when they collide, so...
[tex]\Delta[/tex]U = [tex]\frac{1}{2}[/tex](m1)v[tex]^{2}_{1}[/tex] + [tex]\frac{1}{2}[/tex](m2)v[tex]^{2}_{2}[/tex]

From conservation of momentum, v2 = (m1/m2)v1 so subbing that into the above equation I got:
[tex]\Delta[/tex]U = [tex]\frac{1}{2}[/tex]m1v[tex]^{2}_{1}[/tex] + [tex]\frac{1}{2}[/tex][tex]\frac{m^{2}_{1}}{m_{2}}[/tex]v[tex]^{2}_{1}[/tex]

So then I solved for v1 to get:

v1 = [tex]\sqrt{\frac{2kq_{1}q_{2}((1/(d-r1-r2)-(1/d))}{m_{1}+\frac{m^{2}_{1}}{m_{2}}}}[/tex]

And v2 can be figured out the same way. However, I know for a fact this isn't the right answer.


In closing
I'm don't really know what I did wrong, but I suspect it's because I treated the two spheres as point charges, and I'm not sure if I'm justified in doing that.

faceoclock

Question was solved. Thanks a lot to everyone who took the time to read this :P