Chapter 15, Problem 18A

If a spaceship moves away from you at half the speed of light and fires a rocket away from you at half the speed of light relative to the spaceship, common sense may tell you the rocket moves at the speed of light relative to you. But it doesn’t! The relativistic addition of velocities (not covered in the chapter) is given by $v=\frac{v_1+v_2}{1+\frac{v_1{v}_2}{c^2}}$

Substitute 0.5cfor both $v_1$ and $v_2$ and show that the velocity, v, of the rocket relative to you is 0.8c.

To determine

To show:The velocity $\left(v\right)$ of the rocket relative to an observer is $0.8c$ .

Explanation of Solution

Given:

Velocity of a spaceship is $v_1=0.5c$

Velocity of the rocket relative to the spaceship is $v_2=0.5c$

Formula used:

The relativistic expression for velocity of the rocket is

  $v=\frac{v_1+v_2}{1+\frac{v_1{v}_2}{c^2}}$$\left(1\right)$

Calculation:

Substituting for $v_1$ and $v_2$ in equation $\left(1\right)$ , velocity of the rocket relative to an observer is,

  $v=\frac{0.5c+0.5c}{1+\frac{\left(0.5c\right)\left(0.5c\right)}{c^2}}$

  $=\frac{1.0c}{1+\frac{0.25c^2}{c^2}}$

  $=\frac{1.0c}{1+0.25}$

  $=\frac{1.0c}{1.25}$

  $=0.8c$

Conclusion:

Therefore, the velocity of the rocket relative to an observer is 0. 8c .