(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

The tunnel of the Large Hadron Collider is 27.00 km long. The LHC accelerates protons to a speed of 0.999997828 times the speed of light. How long would it take the proton to pass through the tunnel according to an observer at rest with respect to the building? To an observer moving with the proton?

∆x=27.00 km = 9.00 x 10^{-5}light seconds

v=0.999997848c

2. Relevant equations

Δx′=γ(Δx−vΔt)

Δt′=γ(Δt−vΔx/c2)

L=L_{0}/γ

ΔT=γΔT_{0}

3. The attempt at a solution

For an observer at rest with respect to the building

v_{av}=Δx/Δt

∆t=9.00 x 10^{-5}seconds

I think that part is fine. I tried two different approaches to finding the time in the reference frame of the proton, and got two different answers.

First approach

Treat this as two events:

Event 1) at t=0 and x=0 the proton is at one end of the collider. Then

t' = 0 s and

x' = 0 light seconds

event 2) at t= 9.00 x 10^{-5}seconds and x= 9.00 x 10^{-5}light seconds the proton is at the other end of the collider. Using the Lorentz transformations with v= 0.999997824 c, I get

x'= 9.39 x 10^{-8}light seconds

t'=9.39 x 10^{-8}seconds

So ∆t = 9.30 x 10^{-8}seconds - 0 seconds = 9.30 x 10^{-8}seconds.

Second Approach

From the perspective of the proton, the collider moves towards it at 0.999997824 c. The collider has a rest length of 9.00 x 10^{-5}light seconds, so an observer moving with the proton sees L, where

L=L_{0}/γ = 9.00 x 10^{-5}light seconds * 0.00208614

=1.88 x 10^{-7}light seconds.

A body of that length moves past the observer in time t:

t =L/v= 1.88 x 10^{-7}light seconds/0.999997824 c =1.88 x 10^{-7}seconds.

The second approach is getting me twice the time of the first approach. I also see the same discrepancy in the relativistic lengths. I know I am overlooking something fundamental, but cannot see what.

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# Why don't these methods give me the same answer?

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