Time it takes for an electron and proton to collide

[gsingh2011]

Homework Statement

An electron and positron have mass m_e and separation R and are initially at rest. Find the time T it takes for them to collide.

Homework Equations

F=kq₁q₂/R²

The Attempt at a Solution

My problem is that the force is not constant. Do I integrate the force then? And from what bounds?

 

[vela]

What do you get when you use F=ma?

 

[gsingh2011]

What do you get when you use F=ma?

I don't know, acceleration is changing. Let say we have F=kq₁q₂/r²=ma. Since this problem is symmetrical, I was thinking the two particles would meet in the middle, so each would travel a distance of R/2. So I could divide by m and integrate twice to find the position function, and set that equal to R/2. The only problem with that idea is the force is changing with respect to r, so I can't integrate with respect to t. So I guess that method won't work.

 

[vela]

I don't know, acceleration is changing. Let say we have F=kq₁q₂/r²=ma. Since this problem is symmetrical, I was thinking the two particles would meet in the middle, so each would travel a distance of R/2.

The difference in masses breaks the symmetry. The proton won't move as far as the electron because it's so massive in comparison.

So I could divide by m and integrate twice to find the position function, and set that equal to R/2. The only problem with that idea is the force is changing with respect to r, so I can't integrate with respect to t. So I guess that method won't work.

The standard trick is to multiply by \dot{r}. You'll find it makes both sides of the equation integrable, but first you need to find the correct equation.

Are you familiar with the concept of the reduced mass?

 

[gsingh2011]

The proton won't move as far as the electron because it's so massive in comparison.

It's a positron not a proton, so the mass is the same.

 

[vela]

Oops, not sure why I thought it was a proton. Yes, you're right, they'll meet in the middle.