# Find total E of muon using time dilation

• oddjobmj
In summary, we are given the rest energy and mean lifetime of an Ω- particle, as well as the track length it leaves behind after creation and decay in a particle track detector. Using the equation E=mc2, we can solve for the total energy of the particle. However, to find the velocity of the particle, we must first use time dilation to find the time in the particle's rest frame, and then divide the distance by this time to find the velocity. We are given the proper time t0, and using the equation t=t0\sqrt{1-(\frac{v}{c})^2}, we can solve for the time in the lab frame, t. This can then be used to express
oddjobmj

## Homework Statement

An Ω- particle has rest energy 1672 MeV and mean lifetime 8.2*1011 s. It is created and decays in a particle track detector and leaves a track 24 mm long. What is the total energy of the  particle?

## Homework Equations

E=$\frac{mc^2}{\sqrt{1-(\frac{v}{c})^2}}$

t=t0$\sqrt{1-(\frac{v}{c})^2}$

Rest E=mc2

## The Attempt at a Solution

Because Rest E=mc2 I know that:

E=$\frac{1672 MeV}{\sqrt{1-(\frac{v}{c})^2}}$

What I need to do now is solve for v using the given time and distance (.024 meters). The time is measured in the particle's proper frame so:

t=(8.2*1011 seconds)$\sqrt{1-(\frac{v}{c})^2}$

Of course, this contains v itself. What am I missing?

Thank you!

I am missing what you do with the track length...

oddjobmj said:
Of course, this contains v itself. What am I missing?
Set up an equation for distance in terms of speed and time. All from the view of the lab frame. Then you can solve for v.

Also, I have a rather old particle data book. It claims these ##\Omega^- ## live only ##8.22 \times 10^{-11}## seconds. For our convenience, it also mentions ##c\tau = 0.0246 ## m...

BvU said:
Also, I have a rather old particle data book. It claims these ##\Omega^- ## live only ##8.22 \times 10^{-11}## seconds.
LOL... yeah, the OP contains a typo.

Hah, yes! That would be a rather long time. Unfortunately it is too late to edit the original post.

Also, as Doc Al suggested, I need v to solve this problem. They give me the time and distance it travels in that time but not the velocity. I need to use time dilation to find the time in the rest (station?) frame and divide the distance by that time to find the velocity. I'm just getting mixed up on which t is which and how to solve for t overall.

oddjobmj said:
I'm just getting mixed up on which t is which and how to solve for t overall.
The only time you are given is the proper time (in the muon's frame). What's the time in the lab frame? (Give an expression in terms of v, not a number.)

t=t0$\sqrt{1-(\frac{v}{c})^2}$, where t0 is given

oddjobmj said:
t=t0$\sqrt{1-(\frac{v}{c})^2}$, where t0 is given
Almost. Don't mix up t and t0. (Which must be larger?)

Moving clocks run slower so the time in the particle's proper frame will measure longer than the time in the 'station' frame. In that case t0 needs to be larger but you suggest t is given which flips my assumptions around. Is t0 not the time measured in the particle's proper frame?

t0=$\frac{t}{\sqrt{1-(\frac{v}{c})^2}}$=$\frac{t}{\sqrt{1-(\frac{x}{ct})^2}}$

Should I be using t or t0 inside the square root?

Once I find t0 I can use that in my E calculation to find v?

oddjobmj said:
Moving clocks run slower
Right.

so the time in the particle's proper frame will measure longer than the time in the 'station' frame.
No, just the opposite. The proper time is the shortest time. Every one else observes the particle's 'clock' to run slow. So everyone else measures a greater time.

In that case t0 needs to be larger but you suggest t is given which flips my assumptions around. Is t0 not the time measured in the particle's proper frame?
Yes, t0 is the time measured in the particle's frame.

oddjobmj said:

t0=$\frac{t}{\sqrt{1-(\frac{v}{c})^2}}$=$\frac{t}{\sqrt{1-(\frac{x}{ct})^2}}$
That's not my suggestion. That's just rewriting what you wrote before.

Should I be using t or t0 inside the square root?
Inside the square root there should only be v, not time. You're going to solve for v.

Once I find t0 I can use that in my E calculation to find v?
t0 is given. Once you have the correct expression for t (lab frame time), you'll use it to express distance = speed*time as seen in the lab frame. The only unknown will be v, which you will solve for.

## 1. How does time dilation affect the measurement of the total energy of a muon?

Time dilation, a phenomenon described by Einstein's theory of relativity, states that time moves slower for a moving object compared to a stationary one. This means that the total energy of a muon, which is calculated by multiplying its mass by the square of its velocity, will appear to be greater when measured from a stationary observer's perspective compared to a moving observer's perspective.

## 2. What is the formula for calculating the total energy of a muon using time dilation?

The formula for calculating the total energy of a muon using time dilation is E = γmc2, where γ is the Lorentz factor, m is the mass of the muon, and c is the speed of light.

## 3. Can time dilation be experimentally observed in the measurement of a muon's energy?

Yes, time dilation has been experimentally observed in the measurement of a muon's energy. This was demonstrated in the famous "Muon Lifetime Experiment" at CERN in 1962, where muons were accelerated to high speeds and their lifetimes were measured. The results showed that the muons' lifetimes were extended due to time dilation, confirming Einstein's theory.

## 4. How does the speed of a muon affect its total energy when using time dilation?

According to the formula for total energy using time dilation, the speed of a muon directly affects its total energy. As the muon's speed increases, its Lorentz factor increases, resulting in a higher total energy measurement. This is due to the fact that time dilation causes the muon's perceived mass to increase as it moves faster.

## 5. Are there any other factors that can affect the measurement of a muon's total energy, besides time dilation?

Yes, there are other factors that can affect the measurement of a muon's total energy. These include the muon's initial energy level, any external forces acting on the muon, and the precision of the measurement equipment. However, time dilation is the main factor that must be taken into account when calculating the total energy of a muon.

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