The rogue starship Galaxa is being chased by the...

The rogue starship Galaxa is being chased by the battlecruiser Millenia. The Millenia is catching up to the Galaxa at a rate of $0.55 \mathrm{c}$ 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?

The rogue starship Galaxa is being chased by the battlecruiser Millenia. The Millenia is catching up to the Galaxa at a rate of $0.55 \mathrm{c}$ 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?

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Key Concepts

Invariance of the Speed of Light

One of the foundational principles of special relativity is that the speed of light in a vacuum is constant and the same for all observers, regardless of their state of motion. This concept means that no matter how fast an observer is moving, they will always measure the speed of light as c, and it underpins many of the counterintuitive predictions of relativistic physics.

Relativistic Velocity Addition

In special relativity, velocities do not simply add together as in classical mechanics. Instead, the relativistic velocity addition formula is used to correctly combine velocities when objects approach the speed of light. This mathematical framework ensures that the resulting speed never exceeds the speed of light, maintaining consistency with the invariance of c.

Inertial Reference Frames

The concept of inertial reference frames is essential in understanding measurements in special relativity. An inertial frame is one in which an observer is either at rest or moving at a constant velocity. Observers in different inertial frames may record different values for certain quantities such as time and space intervals, yet the laws of physics, including the invariance of the speed of light, remain the same in all inertial frames.

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