Chapter 15, Problem 20A

Substitute small values of $v_1$ and $v_2$ in the preceding equation and show for everyday speeds that v is practically equal to $v_1+v_2$.

To determine

To show: For everyday speeds, v is practically equal to $v_1+v_2$

Explanation of Solution

Given:

Rocket and the spaceship, both are traveling with smaller speeds almost negligible compared to the speed of light.

Formula used:

According to special theory of relativity, the relative speed is calculated as

  $v=\frac{v_1+v_2}{1+\frac{v_1{v}_2}{c^2}}$

Calculation:

When both rocket and spaceship are traveling with smaller speeds compared to the speed of light then $\frac{v_1{v}_2}{c^2}\ll 1$ . Thus,

  $\begin{array}{l}\frac{v_1{v}_2}{c^2}\approx 0\\ \therefore 1+\frac{v_1{v}_2}{c^2}\approx 1\end{array}$

Therefore,

  $V=\frac{v_1+v_2}{1+0}=v_1+v_2$

Conclusion:

Thus, for smaller rocket and spaceship speeds, $v=v_1+v_2$ .