# Question Clarification involving Muon Velocity and Energy

• Joshk80k
In summary, the question is asking for the average distance traveled by a muon with an energy of 5 GeV and 500 GeV, assuming their speeds are approximately the same. The energy-speed function shows that even small changes in speed near c can result in vastly different energies.
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)

## 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?

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.

## 1. What is a muon?

A muon is a subatomic particle that is similar to an electron, but with a much larger mass. It is classified as a lepton and is found in cosmic rays and in some radioactive decay processes.

## 2. How is the velocity of a muon measured?

The velocity of a muon can be measured using a variety of methods, such as tracking its path through a magnetic field or using time-of-flight measurements. These methods can then be used to calculate the muon's velocity.

## 3. What factors affect the velocity of a muon?

The velocity of a muon can be affected by factors such as its energy, the medium it is traveling through, and any external forces acting upon it. In particular, the energy of a muon has a direct impact on its velocity.

## 4. How does the energy of a muon relate to its velocity?

The energy of a muon is directly proportional to its velocity. This means that as the energy of a muon increases, so does its velocity. This relationship is described by the equation E=γmc², where γ is the Lorentz factor, m is the rest mass of the muon, and c is the speed of light.

## 5. Can a muon travel faster than the speed of light?

No, according to the theory of relativity, the speed of light is the maximum speed at which any particle can travel. This applies to muons as well, so they cannot travel faster than the speed of light. However, they can approach the speed of light, especially at higher energies.

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