Electron and positron length contraction q.

[EngageEngage]

Homework Statement

In the stanford linear accelerator, electrons and positrons are fired at each other at 50Gev. In the lab, each electron and positron beam is 1cm long.
(a) how long is each bundle in its frame?
(b) what is the proper length of the accelerator for a bundle to have both its ends simultaneously in the accelerator in its own reference frame.
(c) what is the length of the positron bundle in the frame of the electron bundle?

Homework Equations

The Attempt at a Solution

I have parts a and b completed, but for c I'm stuck. What I was thinking of doing was finding the relative speeds of the electron and positron and then using this new velocity to compute the new length contraction factor (which would be another gamma). however, it turns out that the speed of each bundle with reference to the accelerator is .999999999947c. Therefore, when I try to find their relative velocities I get the speed of light. does anyone know of another way that I could try doing this?
Thank you for any help

 

[Jhero]

you can provide.
Thank you for your question. To find the length of the positron bundle in the frame of the electron bundle, you can use the Lorentz transformation equations for length contraction. These equations take into account the relative velocity and the Lorentz factor (gamma) between the two frames.

First, we need to find the relative velocity between the two bundles. As you have correctly calculated, the speed of each bundle with reference to the accelerator is 0.999999999947c. However, this is not the same as the relative velocity between the two bundles. To find the relative velocity, we need to use the velocity addition formula in special relativity:

u = (v1 + v2) / (1 + (v1v2)/c^2)

where u is the relative velocity, v1 is the speed of the electron bundle with reference to the accelerator, and v2 is the speed of the positron bundle with reference to the accelerator.

Plugging in the values, we get:

u = (0.999999999947c + 0.999999999947c) / (1 + (0.999999999947c * 0.999999999947c)/c^2)

= (1.999999999894c) / (1 + 0.999999999894)

= 0.999999999947c

This means that the relative velocity between the two bundles is also 0.999999999947c.

Now, we can use the Lorentz transformation equations for length contraction to find the length of the positron bundle in the frame of the electron bundle:

L' = L / gamma

where L' is the contracted length, L is the original length (1cm), and gamma is the Lorentz factor, which is given by:

gamma = 1 / sqrt(1 - (u/c)^2)

Plugging in the values, we get:

gamma = 1 / sqrt(1 - (0.999999999947c / c)^2)

= 1 / sqrt(1 - 0.999999999894)

= 1 / 0.000000000106

= 9.433962264 × 10^9

Now, we can find the contracted length of the positron bundle in the frame of the electron bundle:

L' = (1cm) / (9.433