Question

The mean lifetime of stationary muons is measured to be 2.2000 μs. The mean lifetime of high-speed muons in a burst of cosmic rays observed from Earth is measured to be 15.870 μs. To five significant figures, what is the speed parameter β of these cosmic-ray muons relative to Earth?

Answer 1

Answer:

The speed parameter β of these cosmic-ray is 0.990.

Explanation:

Given that,

Mean lifetime $\Delta t_{0}= 2.2000\ \mu s$

time  $\Delta t=  15.870\ \mu s$

We need to calculate the speed parameter

Using formula of time

$\Delta t=\dfrac{\Delta t_{0}}{\sqrt{1-\dfrac{v^2}{c^2}}}$

Where, $\beta=\sqrt{1-\dfrac{v^2}{c^2}}$

Put the value of $\sqrt{1-\dfrac{v^2}{c^2}}$ into the formula

$\Delta t=\dfrac{\Delta t_{0}}{\sqrt{1-\beta^2}}$

$\dfrac{\Delta t_{0}^2}{\Delta t^2}=1-\beta^2$

$\beta^2=-\dfrac{\Delta t_{0}^2}{\Delta t^2}+1$

Put the value into the formula

$\beta^2=-\dfrac{(2.2000\times10^{-6})^2}{(15.870\times10^{-6})^2}+1$

$\beta=\sqrt{0.9807}$

$\beta=0.990$

Hence,  The speed parameter β of these cosmic-ray is 0.990.