How Do You Calculate the Total Energy of a Particle in Special Relativity?

[bmb2009]

Homework Statement

A particle has a rest energy of 1672MeV and a lifetime of 8.2x10^-11 s. It creates a .024m long track in a lab detector. What is the total energy of the particle

Homework Equations

The Attempt at a Solution

Total Energy = mc^2 + mc^2(1 - A) where A is defined as the Gamma Factor...basically i need to calculate the velocity of the particle and I realize that the lifetime give is the proper time (life of the particle in its frame) and the distance is in the lab frame. So I need to distance/time and I want to convert the proper time to the time interval in the lab frame but how do i do time dilation without the velocity? Thanks

 

[Simon Bridge]

You'll have to find a bunch of relations that have to hold simultaniously - so that you can cancel out v.

 

[bmb2009]

soo...what relations? I'm still stuck

 

[Simon Bridge]

1. total energy
2. time dilation
3. length contraction (related to 2)
4. relationship between distance and time in one reference frame

 

[bmb2009]

got it..thanks

 

[Simon Bridge]

No worries :)

 

[bmb2009]

Ugh nevermind lol I don't have it... and it's driving me crazy.

V= L/T = L'/T' and I know L and T'.

Time dilation says T=T'sqrt(1-A^2) where A is V/c

so i plug in V=L/(T'sqrt(1-A^2)) and solve for V but I get (V^2)(T')^2 - ((V^4)(T')^2)/c^2 = c^2

ahhhhhh

 

[Simon Bridge]

[edit] didn't read all the way ...

Well done.
Sometimes explaining why you have a problem produces the solution.

The particle travels distance L in the lab, in the particle's proper time T, then $L=v\gamma T$
Since the total energy is $E=\gamma E_0$ I have two equations and two unknowns.

When I saw your problem, actually did it via length contraction ... to put everything in the particle's frame.
In that frame, the detector moves a distance $L/\gamma$ in time T - and you get the same equation out.

Notice how I constructed the relations from the physics rather than trying to find the "right" equation to manipulate?