Question Clarification involving Muon Velocity and Energy

[Joshk80k]

Homework Statement

5. What is the average distance traveled by a muon with an energy of 5 GeV? 500 GeV? (assume v is approximately c)

Homework Equations

The Attempt at a Solution

I wasn't really sure where to begin with this. I was under the impression that if in both instances, the velocity is approximately c, they would have the same energy?

Where is the extra 495 GeV coming from in this transition? I'm pretty sure the mass hasn't changed?

Thanks for any help you can offer!

 

[diazona]

The velocity in both cases is approximately c, but not exactly c. The question is not saying that the muons have the same velocity, only that their speeds will both be close to c (and thus close to each other).

Now, think about the function that gives energy in terms of speed:
$$E = \frac{mc^2}{\sqrt{1 - v^2/c^2}}$$
If you've ever seen a graph of this function, you'll know that it has a vertical asymptote at v = c. So for speeds just less than c, the energy rises very quickly with only a small change in the speed. That's how you can have two of the same particle with wildly different energies but nearly the same speed.

Note that if it bothers you to assume that the speed is approximately c, you don't have to. You can solve this problem without making that assumption. But there's one particular place in the calculation where it makes your life easier to assume that v = c (or in other words, to assume the difference between v and c is so small as to be negligible), and the hint is telling you that it's okay to do that.