An interstellar space probe is launched from the Earth....

An interstellar space probe is launched from the Earth. After a brief period of acceleration it moves with a constant velocity, with a magnitude of $70.0 \%$ of the speed of light. Its nuclear-powered batteries supply the energy to keep its data transmitter active continuously. The batteries have a lifetime of 15.0 yr as measured in a rest frame. (a) How long do the batteries on the space probe last as measured by Mission Control on the Earth? (b) How far is the probe from the Earth when its batteries fail as measured by Mission Control? (c) How far is the probe from the Earth when its batteries fail as measured by its built-in trip odometer? (d) For what total time interval after launch are data received from the probe by Mission Control? Note that radio waves travel at the speed of light and fill the space between the probe and the Earth at the time of battery failure.

Key Concepts

Signal Propagation Delay

Signal propagation delay refers to the time it takes for signals, such as radio waves which travel at the speed of light, to traverse the distance between a moving object and an observer. In a relativistic context, even after a moving object stops transmitting data (for example, when its power runs out), data may continue to be received for some additional time due to the finite speed of light. This delay must be accounted for when comparing the times at which events are observed in different frames.

Length Contraction

Length contraction is the relativistic effect that causes the measured length of an object moving relative to an observer to be shorter than its proper length (the length in the object's rest frame). This effect only occurs along the direction of motion and is also governed by the Lorentz factor. It is crucial for understanding differences in distance measurements between observers in different inertial frames.

Time Dilation

Time dilation is the phenomenon where the elapsed time between two events is longer in a frame that is moving relative to the frame in which the events occur at the same location. This means that a clock moving at a high velocity will tick more slowly compared to a clock at rest in an observer’s frame. It is a direct consequence of the Lorentz transformations in special relativity and is used to explain discrepancies such as the extended battery lifetime as observed from different frames.

Lorentz Factor

The Lorentz factor (often denoted by ?) is central to the calculations in special relativity. It quantifies how time intervals, lengths, and relativistic masses transform between different inertial frames moving relative to each other. The factor is defined as ? = 1/?(1 - v²/c²), and it appears in formulas for time dilation and length contraction, determining how much these quantities are scaled due to high relative speeds.

Special Relativity

Special relativity is the framework for understanding the physics of objects moving at speeds close to the speed of light. It introduces effects such as time dilation and length contraction, which become significant at high velocities, and it fundamentally alters the classical concepts of time and space by showing that measurements of time intervals and distances depend on the relative motion of observers.

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