Question

The rogue starship Galaxa is being chased by the battlecruiser Millenia. The Millenia is catching up to the Galaxa at a rate of \(0.55 c\) when the captain of the Millenia decides it is time to fire a missile. First the captain shines a laser range finder to determine the distance to the Galaxa and then he fires a missile that is moving at a speed of \(0.45 c\) with respect to the Millenia. What speed does the Galaxa measure for (a) the laser beam and (b) the missile as they both approach the starship?

Short answer

Answer: The speed of the laser beam measured by the Galaxa is \(c\). The speed of the missile measured by the Galaxa is approximately \(-0.0801c\), where the negative sign indicates that the missile is approaching the Galaxa.

Step-by-step solution

(a) Speed of the laser beam as measured by the Galaxa

Since it is a laser beam, it is a light signal whose speed is always 'c' regardless of the reference frame according to Einstein's theory of special relativity. Therefore, the speed measured by the Galaxa for the laser beam is \(c\).

(b) Speed of the missile as measured by the Galaxa

To find the relative velocity of the missile with respect to the Galaxa, we have to use the relativistic velocity addition formula: \(V_{relative} = \frac{V_{missile} + V_{Galaxa}}{1 + \frac{V_{missile}V_{Galaxa}}{c^2}}\) Here, \(V_{missile} = 0.45c\) and \(V_{Galaxa} = -0.55c\) (negative sign indicates opposite direction). Plugging these values into the formula, we get: \(V_{relative} = \frac{0.45c - 0.55c}{1 - \frac{(0.45c)(-0.55c)}{c^2}}\) After simplifying: \(V_{relative} = \frac{-0.1c}{1 + 0.2475}\) \(V_{relative} = \frac{-0.1c}{1.2475}\) Finally, we get the speed of the missile as measured by the Galaxa: \(V_{relative} = -0.0801c\) (approx.), where the negative sign indicates that the missile is approaching the Galaxa.