Question

Question: A certain particle of mass m has momentum of magnitude mc .What are (a) $\mathit{\beta }$, (b)$\mathit{\gamma }$, and (c) the ratio$\mathit{K}\mathbf{/}{\mathit{E}}_{\mathbf{0}}$?

Short answer

Answer

1. The value of$\beta$is, 0.707.
2. The value of $\gamma$is, 1.414.
3. The value of$\frac{K}{E_o}$ is, 0.414.

Step-by-step solution

Step 1: Identification of the given data

The particle’s momentum magnitude is mc.

Lorentz factor and speed parameter.

The Lorentz factor depends only on velocity and not on the particle’s mass and it is expressed as

$\mathit{\gamma }\mathbf{=}\frac{\mathbf{1}}{\sqrt{\mathbf{1}\mathbf{-}{\mathbf{\beta }}^{\mathbf{2}}}}$

$\mathbf{\beta }$is called the speed parameter which is ratio of speed of particle to speed of light.

.$\mathbf{\beta }\mathbf{}\mathbf{=}\mathbf{}\mathbf{V}\mathbf{/}\mathbf{C}$

Step 3(a and b): Solve the relativistic momentum equation for the given case.

It is given that a particle’s momentum magnitude is mc . The relativistic momentum relation is given by

$p=\gamma mv$

Putting the given momentum in above expression, we get

$$\begin{gathered} mc=\gamma mv \\ \gamma v=c \\ \gamma \frac{v}{c}=1 \\ \gamma \beta =1 \end{gathered}$$ … (1)

We know that the Lorentz factor depends only on velocity and not on the particle’s mass and it is expressed as

$\gamma =\frac{1}{\sqrt{1-{\beta }^2}}$

Inserting this in equation (1), we get

$$\begin{gathered} \frac{\beta }{\sqrt{1-{\beta }^2}}=1 \\ {\beta }^2=1-{\beta }^2 \\ \beta =\frac{1}{\sqrt{2}} \\ =0.707 \end{gathered}$$

Hence the value of $\beta$is, 0.707.

Putting the value of$\beta$ in the equation (1),

$$\begin{gathered} \gamma =\frac{1}{\beta } \\ =\frac{1}{0.707} \\ =1.414 \end{gathered}$$

The value of $\gamma$is, 0.414.

Step 4(c): Determine the ratio K/E0.

The relativistic kinetic energy relation is given by

$$\begin{gathered} K=\left(\gamma -1\right)m{c}^2=\left(\gamma -1\right)E_o \\ \frac{K}{E_o}=\gamma -1 \end{gathered}$$

Substitute all value in the above equation

$$\begin{gathered} \frac{K}{E_o}=1.414- \\ =0.414 \end{gathered}$$

Hence the value of $\frac{K}{E_o}$is,0.414 .

And thus, all the required values are determined.