Question

The half-life of a muon at rest is $1.5\mu s$. Muons that have been accelerated to a very high speed and are then held in a circular storage ring have a half-life of $7.5\mu s$.

a. What is the speed, as a fraction of $c$, of the muons in the storage ring?

b. What is the total energy of a muon in the storage ring? The mass of a muon is $207$times the mass of an electron.

Short answer

a.)$0.980c$

b.)$8.49\times {10}^{-11}J$

Step-by-step solution

Part (a) Step 1: Given Information

We have given that, the half-life of a muon at rest is $1.5\mu s$.

Muons that have been accelerated to a very high speed and are then held in a circular storage ring have a half-life of $7.5\mu s$.

We have to find the speed, as a fraction of $c$, of the muons in the storage ring.

Part (a) Step 2: Simplfy

We know that.

$t^{\text{'}}=\gamma t$

So,

$$\begin{gathered} \gamma =\frac{t^{\text{'}}}{t} \\ =\frac{7.5}{1.5} \\ =5 \end{gathered}$$

Put the form of$\gamma$

$$\begin{gathered} \gamma =\frac{1}{\sqrt{1-{(v/c)}^2}} \\ 5=\frac{1}{\sqrt{1-{(v/c)}^2}} \\ v=\pm 0.980c \end{gathered}$$

The speed will be positive number.

So, the speed of proton is$0.980c$.

Part (b) Step 1: Given Information

We have given that, the half-life of a muon at rest is $1..5\mu s$. Muons that have been accelerated to a very high speed and are then held in a circular storage ring have a half-life of $7.5\mu s$.

We have to find the total energy of a muon in the storage ring.

Part (b) Step 2: Simplify

The total energy of an electron is

$$\begin{gathered} E=\gamma m{c}^2 \\ =5\times 9.11\times {10}^{-31}\times 207\times {(3\times {10}^8)}^2 \\ =8.49\times {10}^{-11}J \end{gathered}$$