Calculating Muon Velocity with Lorentz Transformation | Physics Homework

[kd001]

Homework Statement

Muons, which have a half-life of 2 x 10-6 s, are formed in the Earth's atmosphere at an
altitude of 10 km. If they travel normal to the Earth's surface, and one half of them
reach it before they decay, what is their velocity?

Homework Equations

Lorentz Transormation.

The Attempt at a Solution

My understanding of the question is that in the reference frame of the the muons 2 x 10-6 s elapses before they reach the surface of the Earth. So t' is 2 x 10-6 s. So I need to find out the time that elapses in the Earth's reference frame in order to calculate the relative velocity. However, the only other information I've got is x=10km (taking x to be the direction of the velocity of the muons). Aren't there too many unknowns?

 

[tiny-tim]

Hi kd001!

My understanding of the question is that in the reference frame of the the muons 2 x 10-6 s elapses before they reach the surface of the Earth. So t' is 2 x 10-6 s. So I need to find out the time that elapses in the Earth's reference frame in order to calculate the relative velocity. However, the only other information I've got is x=10km (taking x to be the direction of the velocity of the muons). Aren't there too many unknowns?

Call the velocity v.

Find the time t (in the Earth frame), and then see if t/t' agrees with that velocity.

 

[kd001]

I don't see how t can be calculated without knowing v in the first place.

 

[tiny-tim]

I don't see how t can be calculated without knowing v in the first place.

Find t as a function of v …

what do you get? ​