Question

The proper mean lifetime of a subnuclear particle called a muon is 2 µs. Muons in a beam are traveling at 0.902 c relative to a laboratory. The speed of light is 2.998 × 108 m/s. In the reference frame of the muon, how far does the laboratory travel in a typical lifetime of 2 µs?

Answer 1

To solve this problem it is necessary to apply the equations related to the time measured from a relative viewer and the mathematical equations of motion.

Speed is defined as

$v = \frac{d}{t}$

Where,

d = Distance

t = Time

Re-arrange to find d,

$d = vt \\d = 0.902c*(2*10^{-6})\\d = 0.902(2.998*10^8)*(2*10^{-6})\\d = 540.83m$

Applying the equations of relativity of time,

$t' = \frac{t_0}{\sqrt{1-\frac{v^2}{c^2}}}$

Where,

v = Velocity

c = speed of light

$t_0 =$ Reference time

t = relative time

Replacing,

$t' = \frac{2*10^{-6}}{\sqrt{1-\frac{(0.902c)^2}{c^2}}}$

$t' = \frac{2*10^{-6}}{1-0.902}}$

$t' = 2.04*10^{-5}s$

Therefore the distance would be

$d = vt'\\d = 0.902c*(2.04*10^{-5})\\d = 0.902(2.998*10^8)*(2.04*10^{-5})\\d = 5516.56m$

Therefore the travel will be 5.5Km