A futuristic train moving in a straight line with a uniform...

A futuristic train moving in a straight line with a uniform speed of $0.80 c$ passes a series of communications towers. The spacing between the towers, according to an observer on the ground, is $3.0 \mathrm{~km}$. A passenger on the train uses an accurate stopwatch to see how often a tower passes him. (a) What is the time interval the passenger measures between the passing of one tower and the next? (b) What is the time interval an observer on the ground measures for the train to pass from one tower to the next?

Key Concepts

Special Relativity

Special relativity is the framework that describes the physics of objects moving at high speeds, particularly those approaching the speed of light. It introduces modifications to classical notions of time and space, asserting that measurements of lengths, times, and simultaneity depend on the relative motion between the observer and what is being observed.

Inertial Reference Frames

An inertial reference frame is a frame of reference in which an object either remains at rest or moves at a constant velocity unless acted upon by an external force. Problems like the one described require transforming measurements between different inertial frames — the ground frame and the train frame — using the principles of special relativity.

Time Dilation

Time dilation is the relativistic effect that causes a clock moving relative to an observer to tick slower than a clock at rest in the observer’s frame. This concept is used to explain how the time intervals measured by the passenger on the high-speed train differ from those measured by an observer on the ground.

Length Contraction

Length contraction is the phenomenon in special relativity where the length of an object moving at a high velocity is measured to be shorter along the direction of motion compared to its proper length, which is measured in the object’s rest frame. In this problem, the distance between the towers as observed in the train frame is contracted relative to the ground frame measurement.

Lorentz Factor

The Lorentz factor, often denoted by gamma (?), quantifies the amount of time dilation and length contraction experienced by objects moving at relativistic speeds. It is defined as ? = 1/?(1 ? v²/c²) and plays a critical role in converting measurements between different inertial frames in special relativity.

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