Relativistic Velocity Addition: Spaceship and Missile Scenario

Q6. The captain of spaceship A observes enemy spaceship E escaping with a relative velocity of 0.48c. A missile M is fired from ship A, with a velocity of 0.72c relative to ship A. What is the relative velocity of approach of missile M, observed by the crew on ship E?

Background

Topic: Relativistic Velocity Addition

This question tests your understanding of how to combine velocities in special relativity, especially when objects are moving at speeds close to the speed of light (c). The classical formula for adding velocities does not apply at relativistic speeds; instead, you must use the relativistic velocity addition formula.

Key Terms and Formulas

• c: Speed of light in vacuum

• Relativistic velocity addition formula:

$u' = \frac{u + v}{1 + \frac{uv}{c^2}}$

• u: Velocity of missile relative to spaceship A (0.72c)

• v: Velocity of spaceship A relative to spaceship E (since E is escaping, consider direction carefully)

• u': Velocity of missile relative to spaceship E (what you are solving for)

Step-by-Step Guidance

1. Identify the velocities and their directions. Spaceship E is moving away from A at 0.48c, and the missile is fired from A at 0.72c relative to A.

2. Determine the correct signs for the velocities based on their directions. If both are moving in the same direction, use positive values; if opposite, use negative values.

3. Set up the relativistic velocity addition formula:

   $u' = \frac{u + v}{1 + \frac{uv}{c^2}}$

4. Plug in the values: $u = 0.72c$, $v = 0.48c$ (make sure to check the direction—are both velocities in the same direction relative to E?)

Try solving on your own before revealing the answer!