# Special Relativity and Decaying Particles

1. Mar 6, 2012

### Rapier

1. The problem statement, all variables and given/known data
Pions are part of a class of short-lived particles called mesons which commonly created during the collisions in scientific accelerators or when cosmic rays collide with particles in the upper part of the earth's atmosphere. A type of pion known as a π+ has a mass of 139.6 MeV/c2. One particular π+ was created during a collision in the earth's upper atmosphere and has an total energy E= 1.25 X 105 MeV. As measured in its own frame, it has a life time of 38 nsec before it decays.
a) If it decays at a point 146 meters above sea-level, how high above sea-level was the comsic-ray collision that created the pion?

hcollision = m
255345.052163 NO

HELP: Provide this information in the frame of reference of an observer on earth.
HELP: How are the total energy and mass of an object related?

b) How long does the particle live in the frame of reference of an observer on earth??
Δtearth = nsec

c) In the frame of reference of the pion, how far does it traveled before it decayed?
Δxπ = m *
11.39 OK

2. Relevant equations
d ' = dγ
γ = sqrt (1 - v^2/c^2)
E = K - mc^2
K = mc^2[(1/γ)-1]

3. The attempt at a solution
I used total energy + rest energy (E + mc^2) = kinetic energy (K). When I solved K for V I found that the particle was moving nearly at the speed of light. .999999999004c (that's 9 9s).

I used that speed to calculate my γ = 4.46318e-5.

d' = dγ
11.39m = d (4.46318e-5)
d = 255199.052163 m

The particle decayed 146m about sea level so d+146 = 255345.052163.

But it tells me no, so I believe the error might have come in calculating my V. If I can get A, B will be no problem...but I'm stuck.

Help!! Thanks!

2. Mar 6, 2012

### vela

Staff Emeritus
You have a sign error in one of your equations. The total energy is the sum of the rest energy and the kinetic energy.

Your value for $\gamma$ can't possibly be correct since $\gamma \ge 1$.

3. Mar 7, 2012

### Rapier

I corrected my sign error: E = k + mc^2. However, that doesn't really change anything. My energy is 1.25e5 and the 139.6 MeV of mass isn't making any appreciable change in my kinetic energy. I'm getting the same answer.

4. Mar 8, 2012

### vela

Staff Emeritus
What are you getting for $\gamma$? You should get $\gamma=895$.

Last edited: Mar 8, 2012