# Easy Point Charges question

1. Jan 18, 2013

### f25274

I don't even know why I am having trouble with this problem.

1. The problem statement, all variables and given/known data
Two protons are initially an infinite distance apart. One of the protons is initially at rest, and the other is approaching the other at a certain speed. What kinetic energy must the other proton have so that their minimum separation is again 1 fm?

2. Relevant equations
F=kq1q2/r^2 $\hat{r}$
E=kq/r^2$\hat{r}$
Ei=Ef (No external force)
W=$\Delta$U=∫F*dr=kq1q2/r
3. The attempt at a solution
Assuming that at r=infinity, U=0:
Ki=Uf+2Ke
since the electron will try to repel the other electron, the minimum separation happens when both electrons move at the same speed.
ki=2.304*10^-18J+2Ke

I don't know where to go from here.

2. Jan 18, 2013

### f25274

The answer is 0.46pJ but I don't know why it is that.

3. Jan 19, 2013

### Dick

Work in the center of mass system. If the initial velocity of the proton at infinity is v with zero potential energy then in the center of mass system you have two protons approaching each other with velocity v/2 also at infinity and final kinetic energy at closest approach of both protons is zero.

Last edited: Jan 19, 2013
4. Jan 19, 2013

### haruspex

.. and note that the question says protons, not electrons

5. Jan 19, 2013

### Dick

Good point. But as far as kinetic energy goes it doesn't really matter what they are except for charge and that they are equal mass. The mass should cancel.

6. Jan 19, 2013

### f25274

Haha. Yeah, I forgot that it was a proton. The center of mass way worked! Thanks.
Though I don't get how both their final kinetic energies would be zero. The other proton would start moving as the moving one came closer, right?

Last edited: Jan 19, 2013
7. Jan 19, 2013

### Dick

In the center of mass system they are approaching each other with equal velocities. When they are 1 fm apart both will stop and then move apart. At that point total kinetic energy is zero. Makes it easy.

8. Jan 19, 2013

### f25274

Oh, they aren't moving in opposite directions since the two protons are the same charge.
One is moving in a direction where the other proton is but the other is initially at rest, not moving towards the other proton.

9. Jan 19, 2013

### haruspex

Right, but if you take a frame of reference which is the mass centre of the system at all times then it will move with constant velocity, so constitutes an inertial frame. (It follows from conservation of momentum.) In this frame, the protons will approach each other with equal speed.

10. Jan 19, 2013

### Dick

That's the point to using center of mass coordinates! There one proton has velocity v/2 and the other has velocity -v/2. You can transform back to rest coordinates if you want to but you don't have to.