An observer in a coasting spacecraft moves toward a mirror...

An observer in a coasting spacecraft moves toward a mirror at speed $v=0.650 c$ relative to the reference frame labeled $\mathrm{S}$ in Figure $\mathrm{P} 9.57 .$ The mirror is stationary with respect to $\mathrm{S}$. A light pulse emitted by the spacecraft travels toward the mirror and is reflected back to the spacecraft. The spacecraft is a distance $d=5.66 \times 10^{10} \mathrm{m}$ from the mirror (as measured by observers in $\mathrm{S}$ ) at the moment the light pulse leaves the spacecraft. What is the total travel time of the pulse as measured by observers in (a) the S frame and (b) the spacecraft?

Key Concepts

Relativistic Time Dilation

Time dilation is a relativistic effect that causes time to pass at different rates in different inertial frames. Specifically, a moving clock is observed to tick more slowly compared to a clock at rest in a given inertial frame. This concept is essential when determining how long events, like the round-trip of a light pulse, are perceived by observers in different reference frames.

Lorentz Transformation

The Lorentz transformation equations provide the mathematical framework to relate the coordinates of events (time and space) between different inertial frames. They account for the effects of time dilation and length contraction that occur at high velocities, thus enabling the conversion of measurements from one observer’s frame to another’s in the realm of special relativity.

Relativistic Length Contraction

Length contraction is another outcome of special relativity, where the length of an object in the direction of motion is observed to be shorter in a moving frame compared to its proper length in its rest frame. This effect is significant when comparing spatial measurements across different reference frames, particularly in calculating the apparent distances light must travel in relativistic scenarios.

Invariance of the Speed of Light

One of the cornerstones of Einstein's theory of special relativity is that the speed of light in a vacuum is constant and does not depend on the motion of the observer or the light source. This principle underlies how light propagation and reflections are analyzed in any inertial frame and is key to understanding the behavior of light pulses in different frames of reference.

Inertial Reference Frames

An inertial reference frame is one in which an observer is either at rest or moving at a constant velocity. Observers in different frames can measure different spatial and temporal intervals between the same events. Understanding the concept of inertial frames is crucial when comparing measurements, such as the travel time of a light pulse, as they relate observations between a moving spacecraft and a stationary mirror.

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