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

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

Step-by-step 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=\frac{\Delta t_(0)}{\sqrt{1-(v^2)/(c^2)}}`

Where,
`\beta=\sqrt{1-(v^2)/(c^2)}`

Put the value of
`\sqrt{1-(v^2)/(c^2)}` into the formula

`\Delta t=(\Delta t_(0))/(√(1-\beta^2))`

`(\Delta t_(0)^2)/(\Delta t^2)=1-\beta^2`

`\beta^2=-(\Delta t_(0)^2)/(\Delta t^2)+1`

Put the value into the formula

`\beta^2=-((2.2000*10^(-6))^2)/((15.870*10^(-6))^2)+1`

`\beta=√(0.9807)`

`\beta=0.990`

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