How long after the decay do ##e^{-}## and ##e^{+}## Collide

[Potatochip911]

Homework Statement

A particular type of fundamental particle decays by transforming into an electron $e^{-}$ and a positron $e^{+}$. Suppose the decaying particle is at rest in a uniform magnetic field of magnitude 3.53 mT and the $e^{-}$ and $e^{+}$ move away from the decay point in paths lying in a plane perpendicular to the magnetic field. How long after the decay do the $e^{-}$ and $e^{+}$ collide?

Homework Equations

The Attempt at a Solution

The period which is all we need to solve this problem can be found to be $T=\frac{2\pi m_{e}}{qB}$ for this problem. Personally I think these particles would collide at $t=T$ but in the solutions manual it says they collide at $t=T/2$. This doesn't entirely make sense to me since although both the particles are moving at a speed $v$ they won't have completed the entire rotation at $T/2$ although they will have traveled a total distance $2\pi r$

 

[haruspex]

Maybe I'm missing something, but won't they collide after each has gone half way around the circle?

 

[Potatochip911]

Maybe I'm missing something, but won't they collide after each has gone half way around the circle?

I understand the path they take now. I assumed that both electrons were traveling in the same direction at the start, in this case it will be T before they collide. But if the angle between the two speeds is 180 then it only takes T/2.