Chapter 8, Problem 1P

To determine

To Show: If two particles have equal kinetic energies, the magnitude of their momenta are equal only if they have the same mass.

Explanation of Solution

Formula used:

The kinetic energy of a particle in terms of its momentum is given by

  $K=\frac{p^2}{2m}$

Here, $K$ is the kinetic energy of particle, $p$ is the momentum and $m$ is the mass of particle.

Calculation:

The kinetic energy of two particles is given by

  $K_1=\frac{p_1{}^2}{2m_1}$ and $K_2=\frac{p_2{}^2}{2m_2}$

Equate these kinetic energies as the kinetic energies of two particles are given as equal.

  $\begin{array}{c}{K}_1=K_2\\ \frac{p_1{}^2}{2m_1}=\frac{p_2{}^2}{2m_2}\end{array}$

The magnitude of the momenta of particles are equal.

  $p_1=p_2$

So,

  $\begin{array}{c}\frac{1}{2m_1}=\frac{1}{2m_2}\\ \frac{1}{m_1}=\frac{1}{m_2}\\ m_1=m_2\end{array}$

Conclusion:

Hence, if two particles have equal kinetic energies, the magnitude of their momenta are equal only if they have the same mass.