A spacecraft travels along a straight line from Earth to the...

A spacecraft travels along a straight line from Earth to the Moon, a distance of $3.84 \cdot 10^{8} \mathrm{~m}$. Its speed measured on Earth is $0.50 c$ a) How long does the trip take, according to a clock on Earth? b) How long does the trip take, according to a clock on the spacecraft? c) Determine the distance between Earth and the Moon if it were measured by a person on the spacecraft.

A spacecraft travels along a straight line from Earth to the Moon, a distance of $3.84 \cdot 10^{8} \mathrm{~m}$. Its speed measured on Earth is $0.50 c$
a) How long does the trip take, according to a clock on Earth?
b) How long does the trip take, according to a clock on the spacecraft?
c) Determine the distance between Earth and the Moon if it were measured by a person on the spacecraft.

Key Concepts

Length Contraction

Length contraction is the relativistic phenomenon whereby the length of an object measured parallel to the direction of relative motion appears shorter to an observer in a different inertial frame than to an observer in the object’s rest frame. In the context of a spacecraft traveling at a significant fraction of the speed of light, an observer on board would measure the distance between two points along the direction of travel as being contracted relative to the measurement made by an observer at rest with respect to those two points.

Lorentz Factor

The Lorentz factor, often denoted by ? (gamma), quantifies the amount of time dilation and length contraction that occur at relativistic speeds. It is defined as ? = 1/?(1?(v²/c²)), where v is the relative velocity and c is the speed of light. This factor is used to convert between time intervals and lengths measured in different inertial frames, playing a key role in solving problems involving high-speed travel in special relativity.

Time Dilation

Time dilation is a fundamental concept of special relativity which explains that a clock moving relative to an observer will appear to tick more slowly than a clock at rest with respect to that observer. This effect becomes significant at speeds comparable to the speed of light, and it explains why the duration of an event measured in a moving frame (proper time) is shorter than the duration measured in the stationary frame (coordinate time).

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