A clock moves at a constant velocity of 8.0 km / s with respect to Earth. If the clock ticks at

step 1 Okay, in this video, let's talk about problem number 10. Okay, so the situation here is that we

A clock moves at a constant velocity of 8.0 km / s with...

A clock moves at a constant velocity of $8.0 \mathrm{km} / \mathrm{s}$ with respect to Earth. If the clock ticks at intervals of one second in its rest frame, how much more than a second elapses between ticks of the clock as measured by an observer at rest on Earth? [Hint: Use the binomial approximation.

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Key Concepts

Binomial Approximation

The binomial approximation is used to simplify expressions like the Lorentz factor when v << c. It approximates ? as approximately 1 + ½(v²/c²), allowing for an easier calculation of small corrections to the time interval without resorting to the exact expression for high precisions.

Time Dilation

Time dilation is a phenomenon in special relativity where the elapsed time between events is longer when measured in a frame moving relative to the clock's rest frame. This effect means that a clock moving relative to an observer ticks more slowly, so that the interval between ticks appears longer than one second in the observer’s frame.

Proper Time and Coordinate Time

Proper time is the time interval measured by a clock at rest relative to the event being timed, while coordinate time is measured in a frame where the clock is in motion. The difference between these times is used to quantify time dilation in special relativity.

Lorentz Factor

The Lorentz factor, denoted by gamma (?), quantifies the amount of time dilation and length contraction. It is defined as ? = 1/?(1 – v²/c²), where v is the relative velocity and c is the speed of light. For objects moving at speeds much less than c, ? is very close to 1, resulting in minimal time dilation.

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