Proton-Proton Interaction at Minimum Separation

[f25274]

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

Homework Statement

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?

Homework Equations

F=kq₁q₂/r^2 \hat{r}
E=kq/r^2\hat{r}
E_i=E_f (No external force)
W=\DeltaU=∫F*dr=kq₁q₂/r

The Attempt at a Solution

Assuming that at r=infinity, U=0:
K_i=U_f+2K_e
since the electron will try to repel the other electron, the minimum separation happens when both electrons move at the same speed.
k_i=2.304*10^-18J+2K_e

I don't know where to go from here.

 

[f25274]

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

 

[Dick]

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

Homework Statement

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?

Homework Equations

F=kq₁q₂/r^2 \hat{r}
E=kq/r^2\hat{r}
E_i=E_f (No external force)
W=\DeltaU=∫F*dr=kq₁q₂/r

The Attempt at a Solution

Assuming that at r=infinity, U=0:
K_i=U_f+2K_e
since the electron will try to repel the other electron, the minimum separation happens when both electrons move at the same speed.
k_i=2.304*10^-18J+2K_e

I don't know where to go from here.

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.

 

[haruspex]

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

 

[Dick]

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

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.

 

[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?

 

[Dick]

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 electron would start moving as the moving one came closer, right?

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.

 

[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.

 

[haruspex]

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.

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.

 

[Dick]

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.

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.