Special Relativity and Decaying Particles

[Rapier]

Homework Statement

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

Homework Equations

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

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!

 

[vela]

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 $\gamma \ge 1$.

 

[Rapier]

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 $\gamma \ge 1$.

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.

 

[vela]

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

 

[asteriatic]

I can confirm that your calculations for part (a) are correct. The error may have come from rounding off your calculated speed, which could have affected your calculated value for γ. Try using more significant figures in your calculations to see if that makes a difference. Additionally, make sure that your units are consistent throughout your calculations.

For part (b), the particle's lifetime in the frame of reference of an observer on Earth can be calculated using the formula Δtearth = γΔtπ, where Δtπ is the lifetime of the pion in its own frame of reference. Using the value for γ that you calculated in part (a), you can find the particle's lifetime in the Earth frame of reference.

For part (c), the distance traveled in the pion's own frame of reference can be calculated using the formula Δxπ = vΔtπ, where v is the speed of the pion as calculated in part (a). This will give you the distance traveled before the pion decays.

Remember to check your units and use appropriate significant figures throughout your calculations. Keep up the good work!