A spacecraft moving at 0.95 c travels from the Earth to the...

A spacecraft moving at $0.95 \mathrm{c}$ travels from the Earth to the star Alpha Centauri, which is $4.5$ light years away. How long will the trip take according to (a) Earth clocks and $(b)$ spacecraft clocks? (c) How far is it from Earth to the star according to spacecraft occupants? $(d)$ What do they compute their speed to be?

A spacecraft moving at $0.95 \mathrm{c}$ travels from the Earth to the star Alpha Centauri, which is $4.5$ light years away. How long will the trip take according to (a) Earth clocks and $(b)$ spacecraft clocks?
(c) How far is it from Earth to the star according to spacecraft occupants? $(d)$ What do they compute their speed to be?

Key Concepts

Inertial Reference Frames

Inertial reference frames are coordinate systems in which objects either remain at rest or move at a constant velocity unless acted upon by external forces. In the context of this problem, the Earth and the spacecraft each serve as inertial frames, and the differences in measurements of time and distance arise because of their relative motion at speeds approaching the speed of light.

Proper Time

Proper time is the time interval measured by an observer who is at rest relative to the events being observed—in this case, the time measured by the clocks aboard the spacecraft. It represents the actual experienced time for the moving observers and is shorter than the coordinate time measured in another inertial frame, such as that of Earth.

Lorentz Factor

The Lorentz factor, denoted by gamma (?), quantifies the amount of time dilation and length contraction experienced by an object moving at relativistic speeds. It is calculated as ? = 1/?(1 - v²/c²), where v is the relative speed and c is the speed of light. This factor is central to adjusting measurements of time and distance between different inertial frames.

Length Contraction

Length contraction refers to the phenomenon where an observer in motion relative to an object measures the length of that object (or distance between two points aligned with the direction of motion) to be shorter than an observer at rest relative to the object. This concept explains how passengers in the spacecraft perceive the distance to Alpha Centauri as less than what an Earth observer measures.

Time Dilation

Time dilation is a key concept in special relativity stating that time passes at different rates in different inertial frames, especially at speeds close to the speed of light. This means that a clock moving at a high velocity will tick slower relative to a clock at rest, which explains why the spacecraft’s clock records a shorter time than the Earth clocks for the same journey.

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