Question

The positive muon (?+), an unstable particle, lives on average 2.20×10?6s (measured in its own frame of reference) before decaying. a. If such a particle is moving, with respect to the laboratory, with a speed of 0.910 c , what average lifetime is measured in the laboratory? b. What average distance, measured in the laboratory, does the particle move before decaying?

Answer 1

Answer:

$4.82383\times 10^{-6}\ s$

1316.90559 m

Explanation:

$\Delta T'$ = Time measured in the laboratory

$\Delta T$ = Time measured in its own frame of reference = $2.2\times 10^{-6}\ s$

c = Speed of light = $3\times 10^8\ m/s$

v = 0.91 c

Time dilation is given by

$\Delta T'=\dfrac{\Delta T}{\sqrt{1-\dfrac{v^2}{c^2}}}\\\Rightarrow \Delta T'=\dfrac{2\times 10^{-6}}{\sqrt{1-\dfrac{0.910^2c^2}{c^2}}}\\\Rightarrow \Delta T'=4.82383\times 10^{-6}\ s$

The average lifetime is measured in the laboratory is $4.82383\times 10^{-6}\ s$

Distance measured would be

$L=v\Delta T'\\\Rightarrow L=0.91\times 3\times 10^8\times 4.82383\times 10^{-6}\\\Rightarrow L=1316.90559\ m$

The distance is 1316.90559 m