An unstable particle is created in the upper atmosphere from a cosmic ray and travels straight down toward the surface of the earth with a speed of 0.99537 c relative to the earth. A scientist at rest on the earth's surface measures that the particle is created at an altitude of 40.0 km .

QUESTION 1: As measured by the scientist, how much time does it take the particle to travel the 40.0 km to the surface of the earth?
QUESTION 2: Use the length-contraction formula to calculate the distance from where the particle is created to the surfaceof the earth as measured in the particle's frame.
QUESTION 3: In the particle's frame, how much time does it take the particle to travel from where it is created to the surface of the earth?

Q1)  Time taking by particle to travel the 40 km wrt. earth = sec.

Q2) The distance traveled by particle where the particle created to surface of earth wrt. particle's frame = 3.84 km.

Q3) The time taking by particle to travel from where it is created to the surface of the earth = sec.

Explanation:

Given :

Speed of particle wrt. earth

Distance between where particle is created and earth surface = 40 km

we know that,

⇒      

Where , , we know speed of light

∴      

∴ Thus, time taking by particle to travel the 40 km wrt. earth

According to the lorentz transformation,

⇒    

Where improper length, proper length (distance measured wrt. rest frame) = 40 km

      km

∴ Thus, the distance traveled by particle where the particle created to surface of earth wrt. particle's frame = 3.84 km.

According to the time dilation,

Where improper time (wrt. earth frame time) ,   proper time (wrt. particle frame).

Thus, the time taking by particle to travel from where it is created to the surface of the earth .