A rocket cruising past earth at 0.800 c shoots a bullet out...

A rocket cruising past earth at $0.800 c$ shoots a bullet out the back door, opposite the rocket's motion, at $0.900 \mathrm{c}$ relative to the rocket. What is the bullet's speed relative to the earth?

Key Concepts

Special Relativity

Special relativity is a theory proposed by Einstein that redefines how space and time are interrelated for observers in different inertial frames. It introduces the idea that the laws of physics are the same for all non-accelerating observers and that the speed of light is constant in all inertial frames, which leads to phenomena such as time dilation and length contraction.

Inertial Frames of Reference

Inertial frames of reference are coordinate systems in which an object either is at rest or moves at a constant velocity. When analyzing problems involving objects moving at high speeds, it is essential to specify different inertial frames (such as that of the rocket or the Earth) and accurately transform measurements like velocity between these frames.

Relativistic Velocity Addition

Relativistic velocity addition is a formula that replaces the classical notion of simply adding velocities, ensuring that the resultant speed never exceeds the speed of light. This formula takes into account the effects of time dilation and length contraction that occur at speeds close to the speed of light, and it is critical for correctly calculating the relative speeds between objects in different inertial frames.

Speed of Light Invariance

The speed of light invariance is a fundamental postulate of special relativity stating that the speed of light in a vacuum is constant and independent of the motion of the light source or observer. This principle underpins the modifications to classical mechanics at high velocities and directly impacts how velocities are added in relativistic contexts.

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