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

step 1 So the question was asking us how much more time will be measured on Earth? Well, we know the cl

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 $\left.\left(1-v^{2} / c^{2}\right)^{-1 / 2} \approx 1+v^{2} /\left(2 c^{2}\right) .\right]$

Key Concepts

Time Dilation

Time dilation is a key prediction of special relativity which states that a clock moving relative to an observer will appear to tick slower than a clock at rest with respect to that observer. This effect arises because the laws of physics, including the measurement of time intervals, are relative to the inertial reference frame and become distorted when relative motion at high speeds is involved.

Proper Time

Proper time refers to the time interval measured by an observer who is at rest relative to the events being timed. It is considered the true measure of time in the object's own frame and is always the shortest measured interval, with any other observer in relative motion recording a longer interval due to time dilation.

Lorentz Factor

The Lorentz factor, often denoted as gamma, is a crucial component in calculations related to time dilation and length contraction in special relativity. It is defined as 1/sqrt(1 - v²/c²) and quantifies the amount by which time, length, and relativistic mass change for an object moving at a certain velocity relative to an observer.

Binomial Approximation

The binomial approximation is a mathematical technique used to simplify calculations when dealing with small values relative to one. In the context of time dilation, it allows us to approximate the Lorentz factor for speeds much smaller than the speed of light, by expressing (1 - v²/c²)?¹/² as approximately 1 + v²/(2c²), thereby making analytical calculations more manageable.

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