In the age of interstellar travel, an expedition is mounted...

In the age of interstellar travel, an expedition is mounted to an interesting star 2000.0 light-years from Earth. To make it possible to get volunteers for the expedition, the planners guarantee that the round-trip to the star will take no more than $10.000 \%$ of a normal human lifetime. (At the time, the normal human lifetime is 400.00 years.) What is the minimum speed with which the ship carrying the expedition must travel?

Key Concepts

Time Dilation

Time dilation is the effect predicted by special relativity where a clock moving at a high speed relative to an observer ticks more slowly than one at rest relative to that observer. This concept is critical in scenarios such as interstellar travel since the proper time experienced by the travelers is less than the time measured by an observer at rest (e.g., on Earth), allowing journeys that seem extremely long from Earth’s perspective to be experienced as much shorter by the travelers.

Special Relativity

Special relativity is a theory formulated by Einstein that describes how the measurements of space and time change for observers in different inertial frames, especially at speeds close to the speed of light. A key consequence is that time intervals and distances measured in the moving frame (proper time and contracted length) differ from those measured in the rest frame, which is essential for problems involving interstellar travel where significant fractions of the speed of light are involved.

Lorentz Factor

The Lorentz factor, commonly denoted by gamma (?), quantifies the amount of time dilation (and length contraction) experienced at relativistic speeds. It is defined as 1/sqrt(1 - v^2/c^2) where v is the velocity of the moving frame and c is the speed of light. This factor is used to relate the proper time experienced by the travelers to the coordinate time measured in the rest frame, and is fundamental in obtaining the minimum travel speed required for the journey to fall within a given time constraint.

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