Chapter 6, Problem 1Q

To determine

To show that the units of $\frac{p^2}{2m}$ and $m{c}^2$ are joules.

Answer to Problem 1Q

It is showed that the units of $\frac{p^2}{2m}$ and $m{c}^2$ are joules.

Explanation of Solution

Write the equation for momentum of a particle.

  $p=mv$        (I)

Here, $p$ is the momentum, $m$ is the mass of the particle and $v$ is the speed of the particle.

Put the above equation in $\frac{p^2}{2m}$ .

  $\begin{array}{c}\frac{p^2}{2m}=\frac{{\left(mv\right)}^2}{2m}\\ =\frac{1}{2}m{v}^2\end{array}$

The unit of $m$ is kg and the unit of $v$ is m/s.

Replace the variables in the above equation by corresponding units.

  $\begin{array}{c}\frac{p^2}{2m}=\left(\text{kg}\right){\left(\text{m/s}\right)}^2\\ =\left(\text{kg}\cdot {\text{m/s}}^2\right)\left(\text{m}\right)\\ =\text{N}\cdot \text{m}\\ =\text{J}\end{array}$

The unit of $c$ is m/s.

Replace the variables in $m{c}^2$ by the corresponding units.

  $\begin{array}{c}m{c}^2=\left(\text{kg}\right){\left(\text{m/s}\right)}^2\\ =\left(\text{kg}\cdot {\text{m/s}}^2\right)\left(\text{m}\right)\\ =\text{N}\cdot \text{m}\\ =\text{J}\end{array}$

Conclusion:

Thus, it is showed that the units of $\frac{p^2}{2m}$ and $m{c}^2$ are joules.