An observer in a rocket moves toward a mirror at speed v...

An observer in a rocket moves toward a mirror at speed v relative to the reference frame labeled by S in Figure P39.65. The mirror is stationary with respect to S. A light pulse emitted by the rocket travels toward the mirror and is reflected back to the rocket. The front of the rocket is a distance $d$ from the mirror (as measured by observers in S) at the moment the light pulse leaves the rocket. What is the total travel time of the pulse as measured by observers in (a) the $S$ frame and (b) the front of the rocket?

Key Concepts

Length Contraction

Length contraction refers to the phenomenon where the length of an object measured along the direction of its motion is shorter in the frame in which it is moving compared to its rest frame. This concept is important when analyzing distances between objects (like the rocket and the mirror) in different frames, as the contracted distance influences time calculations for moving light pulses.

Inertial Frames of Reference

In special relativity, inertial frames are reference systems in which objects move at constant velocity, and the laws of physics have the same form. Analyzing events from different inertial frames—such as a rocket’s rest frame versus a stationary frame (S)—highlights how the measurements of time and distance can differ while still adhering to the principle of relativity.

Invariance of the Speed of Light

A cornerstone of special relativity is that the speed of light in vacuum is constant and equal to c in every inertial frame, regardless of the motion of the source or observer. This principle underlies the analysis of light travel times, ensuring that, despite the relative motion between the rocket and the mirror, light always propagates at the same speed in all calculations.

Proper Time and Coordinate Time

Proper time is the time interval measured by an observer for whom the events occur at the same spatial position, such as timing by a clock at rest relative to the event. In contrast, coordinate time is measured in a frame where the events occur at different locations. In problems involving moving observers like the rocket, distinguishing between these two helps in understanding the differing time measurements in the rocket’s frame versus the stationary frame S.

Time Dilation

Time dilation is the effect in special relativity where a moving clock is observed to tick slower compared to a clock at rest. It explains why the time interval (such as the round-trip travel time of a light pulse) measured in one inertial frame (S) will differ from the proper time measured in the moving frame (rocket).

Lorentz Transformations

Lorentz transformations provide the mathematical framework for converting space and time coordinates from one inertial frame to another moving at a constant velocity relative to the first. They ensure that the fundamental relationships—such as the invariance of the speed of light—are maintained, and they are essential tools for analyzing the differences in time measurements between observers in the rocket frame and the S frame.

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