Question

Suppose you are explaining the theory of relativity to a friend, and you tell him that nothing can go faster than \(300,000 \mathrm{~km} / \mathrm{s}\). He says that is obviously false: Suppose a spaceship traveling past you at \(200,000 \mathrm{~km} / \mathrm{s}\), which is perfectly possible according to what you are saying, fires a torpedo straight ahead whose speed is \(200,000 \mathrm{~km} / \mathrm{s}\) relative to the spaceship, which is also perfectly possible; then, he says, the torpedo's speed is \(400,000 \mathrm{~km} / \mathrm{s}\) How would you answer him?

Short answer

Answer: The actual velocity of the torpedo with respect to a stationary observer is approximately 88,021 km/s.

Step-by-step solution

Understand the problem

According to the theory of relativity, no object can travel faster than the speed of light, which is approximately \(300,000 \mathrm{km/s}\). The scenario presented states that a spaceship is traveling at a speed, \(u_{ship} = 200,000 \mathrm{~km/s}\), and it fires a torpedo with a speed, \(u_{torpedo} = 200,000\mathrm{~km/s}\), relative to the spaceship. We need to find the actual velocity of the torpedo relative to an observer who is not moving along with the spaceship.

Relativistic addition of velocities

To find the velocity of the torpedo with respect to an observer not moving along with the spaceship, we cannot simply add the velocities of the spaceship and the torpedo. Instead, we must use the relativistic velocity addition formula, which states: $$ v_{total} = \frac{u_{ship} + u_{torpedo}}{1 + \frac{u_{ship}\cdot u_{torpedo}}{c^2}} $$ where \(v_{total}\) is the total velocity of the torpedo relative to the stationary observer, \(u_{ship}\) is the velocity of the spaceship, \(u_{torpedo}\) is the velocity of the torpedo relative to the spaceship, and \(c\) is the speed of light.

Calculate the total velocity

Now, we can plug in the values for \(u_{ship}\), \(u_{torpedo}\), and \(c\) to find the total velocity of the torpedo with respect to the stationary observer: $$ v_{total} = \frac{200,000 + 200,000}{1 + \frac{200,000\cdot200,000}{300,000^2}} = \frac{400,000}{1 + \frac{40000}{9}} = \frac{400,000}{1+\frac{4000}{90}} = \frac{400,000}{\frac{4090}{90}} $$ $$ v_{total} = \frac{400,000 \cdot 90}{4090} = \frac{36,000,000}{409} \approx 88,021 \mathrm{~km/s} $$

Conclusion

The relativistic velocity addition formula shows that the total velocity of the torpedo with respect to a stationary observer is approximately \(88,021\mathrm{~km/s}\). This result is consistent with the theory of relativity, as the calculated velocity is less than the speed of light (\(300,000 \mathrm{~km/s}\)). Therefore, we can explain to our friend that their initial intuition is incorrect because the velocities should be added using the relativistic velocity addition formula rather than simple linear addition.