How Close to Light Speed Do Protons Move in the LHC?

[Orodruin]

n the first method you used the approximation 1−ϵ1−ϵ1-\epsilon in the calculations for part b) and dropped the term in ϵ2ϵ2\epsilon^2 (which wasn't needed for part a). If you put that term back in, you should get the same answer as in the second method.

This was never an issue. The relation $v=c(1-\epsilon)$ from (a) is what defines $\epsilon$ and it is therefore exact. He only needs to keep terms to order $\epsilon^2$ when adding the velocities, which he did.

 

[spaghetti3451]

This was never an issue. The relation $v=c(1-\epsilon)$ from (a) is what defines $\epsilon$ and it is therefore exact. He only needs to keep terms to order $\epsilon^2$ when adding the velocities, which he did.

I thought the same.

 

[spaghetti3451]

Which is the order of $\epsilon^2$.

In the first method you used the approximation $1-\epsilon$ in the calculations for part b) and dropped the term in $\epsilon^2$ (which wasn't needed for part a). If you put that term back in, you should get the same answer as in the second method.

Ignored this comment.

You may also want to check that value for $\epsilon$. Shouldn't it be approx $10^{-8}$?

Showed the equivalence of the two approaches using this piece of advice.

 

[spaghetti3451]

Thanks so much to vela, Orodruin and PeroK for helping me out with this problem with your suggestions and problem-solving tips!

I've learned a lot from solving just this one single problem.