Question

To eight significant figures, what is speed parameter if the Lorentz factor $\gamma$ is (a) 1.010 000 0, (b) 10.000 000, (c) 100.000 00, and (d) 1000.000 0?

Short answer

1. The speed parameter is $0.14037076$.
2. The speed parameter is $0.99498743$.
3. The speed parameter is $0.99994999$.
4. The speed parameter is $0.99999950$.

Step-by-step solution

Identification of given data 

The Lorentz factor for speed parameter is ${\gamma }_a=1.0100000$

The Lorentz factor for speed parameter is ${\gamma }_b=10.000000$

The Lorentz factor for speed parameter is ${\gamma }_c=100.00000$

The Lorentz factor for speed parameter is ${\gamma }_d=1000.0000$

The Lorentz factor is used to find the speed parameter which is the ratio of the speed of particle with speed of light.

Determination of speed parameter  (a)

The speed factor of muon is given as:

$\beta =\sqrt{1-{\left(\frac{1}{{\gamma }_a}\right)}^2}$

Substitute all the values in equation.

$$\begin{gathered} \beta =\sqrt{1-{\left(\frac{1}{1.0100000}\right)}^2} \\ \beta =0.14037076 \end{gathered}$$

Therefore, the speed parameter is $0.14037076$.

Determination of speed parameter  (b)

The speed factor of muon is given as:

$\beta =\sqrt{1-{\left(\frac{1}{{\gamma }_b}\right)}^2}$

Substitute all the values in equation.

$\beta =\sqrt{1-{\left(\frac{1}{10.000000}\right)}^2}$

$\beta =0.99498743$

Therefore, the speed parameter is $0.99498743$.

Determination of speed parameter  (c) 

The speed factor of muon is given as:

$\beta =\sqrt{1-{\left(\frac{1}{{\gamma }_c}\right)}^2}$

Substitute all the values in equation.

$$\begin{gathered} \beta =\sqrt{1-{\left(\frac{1}{100.00000}\right)}^2} \\ \beta =0.99994999 \end{gathered}$$

Therefore, the speed parameter is $0.99994999$.

Determination of speed parameter  (d)

The speed factor of muon is given as:

$\beta =\sqrt{1-{\left(\frac{1}{{\gamma }_d}\right)}^2}$

Substitute all the values in equation.

$$\begin{gathered} \beta =\sqrt{1-{\left(\frac{1}{1000.0000}\right)}^2} \\ \beta =0.99999950 \end{gathered}$$

Therefore, the speed parameter is $0.99999950$.