Because of the earth's rotation, a person living on top of a...

Because of the earth's rotation, a person living on top of a mountain moves at a faster speed than someone at sea level. The mountain dweller's clocks thus run slowly compared to those at sea level. If the average life span of a hermit is 80 years, on average how much longer would a hermit dwelling on the top of a $3000-\mathrm{m}-$ high mountain live compared to a sea-level hermit?

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Key Concepts

Rotational Kinematics and Tangential Velocity

Because the Earth rotates, a point on its surface moves with a tangential speed that increases with the distance from the axis of rotation. Individuals at higher altitudes on Earth, such as on a mountain, are farther from the axis and therefore travel at a greater linear speed compared to those at sea level, which leads to slight differences in clock rates according to relativity.

Special Relativity (Time Dilation)

This concept explains that a clock in motion relative to an inertial observer will tick more slowly than one at rest. It is a fundamental result of Einstein’s theory of special relativity that applies even to small differences in speed, such as those caused by Earth’s rotation, though the effect becomes noticeable only at speeds close to that of light.

Low-Speed Approximation in Relativity

When dealing with speeds that are much lower than the speed of light, the time dilation effect can be approximated by a series expansion where the dominant term is proportional to the square of the speed divided by the square of the speed of light. This approximation is useful for estimating how small differences in velocity (such as those due to altitude differences) affect the passage of time over long time periods.

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