Chapter 26, Problem 64GP

(a)

To determine

The speed of electron.

Answer to Problem 64GP

  $\text{0}\text{.986}\text{c}$

Explanation of Solution

Given info:

Kinetic energy of electron is $\text{5}\text{.00}$ times its rest mass.

Formula used:

  $m=\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}$

Where,

  $m_0$ is rest mass

  $m$ is relativistic mass

  $v$ is velocity

  $c$ is speed of light, $3.00\times {10}^{8}m/s$

Calculation:

If the kinetic energy is $5.00$ times as great as the rest energy, then:

  $\begin{array}{l}KE=\left(m-m_0\right)c^2=5.00m_0{c}^2\\ \Rightarrow m=6.00m_0\end{array}$

Now, calculate the speed by using $m=\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}$

  $6.00m_0=\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}$

  $v=0.986c$

(b)

To determine

The speed of the electron.

Answer to Problem 64GP

  $0.999c$

Explanation of Solution

Given info:

The kinetic energy of the electron is 999 times as great as the rest energy.

Formula used:

  $m=\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}$

Where,

  $m_0$ is rest mass

  $m$ is relativistic mass

  $v$ is velocity

  $c$ is speed of light, $3.00\times {10}^{8}m/s$

Calculation:

If the kinetic energy of the electron is 999 times as great as the rest energy, then:

  $\begin{array}{l}KE=\left(m-m_0\right)c^2=999m_0{c}^2,\\ \Rightarrow m=1000m_0\end{array}$

Now, calculate the speed by using $m=\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}$

  $1000m_0=\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}$

  $v=0.999c$