You fly 5000 km across the United States on an airliner at...

You fly $5000 \mathrm{km}$ across the United States on an airliner at $250 \mathrm{m} / \mathrm{s} .$ You return two days later at the same speed.
a. Have you aged more or less than your friends at home?
b. By how much?
Hint: Use the binomial approximation.

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Key Concepts

Proper Time

Proper time refers to the time interval measured by a clock that travels along with an observer. It is the actual time experienced by that observer, which, due to relativistic effects such as time dilation, may differ from the time measured by an observer in a different inertial frame.

Binomial Approximation

The binomial approximation is a mathematical method used to simplify calculations involving expressions raised to a power when dealing with small quantities. In the context of time dilation, it allows the Lorentz factor to be approximated when the velocity is much smaller than the speed of light (v << c), making it easier to calculate the slight difference in elapsed time.

Lorentz Factor

The Lorentz factor, usually denoted by ? (gamma), is a measure of how much time, length, and relativistic mass change for an object while moving relative to an observer. It is defined as 1/?(1 - v²/c²) and is directly used to quantify time dilation effects between moving and stationary frames.

Time Dilation

Time dilation is a core phenomenon of special relativity where a moving clock, as seen from a stationary frame, ticks slower than one at rest. This effect means that a traveler moving at high speeds experiences less elapsed time compared to someone who remains stationary, which is the basis of the twin paradox scenario.

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