# Find total E of muon using time dilation

**1. The problem statement, all variables and given/known data**

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?

**2. Relevant equations**

E=[itex]\frac{mc^2}{\sqrt{1-(\frac{v}{c})^2}}[/itex]

t=t_{0}[itex]\sqrt{1-(\frac{v}{c})^2}[/itex]

Rest E=mc^{2}

**3. The attempt at a solution**

Because Rest E=mc^{2} I know that:

E=[itex]\frac{1672 MeV}{\sqrt{1-(\frac{v}{c})^2}}[/itex]

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)[itex]\sqrt{1-(\frac{v}{c})^2}[/itex]

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

Thank you!

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