Question

A clock moves along the \(x\) axis at a speed of \(0.622 c\) and reads zero as it passes the origin. (a) Calculate the Lorentz factor. (b) What time does the clock read as it passes \(x=183 \mathrm{~m}\) ?

Short answer

The Lorentz factor (\(\gamma\)) is approximately 1.301, and the time read by the moving clock as it passes \(x=183m\) is approximately \(7.09 \times 10^{-7}\) seconds.

Step-by-step solution

Calculate the Lorentz factor

The Lorentz factor is given by the formula \(\gamma = \frac{1}{\sqrt{1-(v/c)^2}}\), where \(v\) is the velocity of the moving observer, and \(c\) is the speed of light. The problem provides the speed of the clock in terms of \(c\) as \(v = 0.622c\). Substituting the given values in the formula, the Lorentz factor \(\gamma\) can be calculated.

Calculate the time read by the moving clock

In special relativity, the time read by the moving clock under the influence of time dilation is given by the formula \(t'=\gamma t\), where \(t'\) is the dilated time, \(\gamma\) is the Lorentz factor, and \(t\) is the proper time. In our case, the clock reads zero as it passes the origin. This means \(t=0\). The only time it experiences is a result of its motion. Therefore, the proper time \(t\) can be written as the distance to travel divided by the speed of light: \(t=x/c\). Now substitute \(x=183m\) into \(t=x/c\) and we get the proper time. The dilated time \(t'\) can be calculated by multiplying the Lorentz factor \(\gamma\) with the proper time \(t\).

Key concepts

Special Relativity

Understanding special relativity is like stepping into a world where the rules we're used to no longer apply. Instead, we enter a universe where time and space intertwine in an intricate dance. This groundbreaking theory, introduced by Albert Einstein in 1905, revolutionized physics by showing that the laws of motion and the speed of light remain constant for all observers, regardless of their velocity.

When we talk about objects moving at a significant fraction of the speed of light, we enter the realm of special relativity. It teaches us that as an object's speed increases, its length contracts and its time dilates when viewed from a different frame of reference. These effects are hard to notice at everyday speeds but become dramatically significant as you approach the speed of light. At the heart of these phenomenal changes lies the Lorentz factor, a quantity that lets us calculate how time stretches and lengths contract in this relativistic universe.

Time Dilation

Time dilation is a mind-bending concept that sounds like science fiction but is very much a part of the real laws of physics. According to special relativity, a moving clock will tick slower than a stationary one. This isn't due to any fault in the clock's mechanism but rather how time itself behaves at high velocities.

Let's make this abstract idea more concrete with a thought experiment. Imagine two twins are separated, with one travelling in a spacecraft at a significant fraction of the speed of light while the other stays on Earth. Upon the space-faring twin's return, they will find that they have aged less than their sibling that remained behind. This effect, predicted by special relativity, has been confirmed through experiments involving high-speed particles and precision clocks aboard spacecraft.

The exercise that students are working on illustrates time dilation. By calculating the Lorentz factor, they can determine how much the clock's reading is affected as it moves at speeds comparable to the speed of light. Therefore, time dilation is not just a quirk of mathematical equations, but a tangible phenomenon that has real-world implications, especially in high-speed travel and the synchronization of clocks in GPS satellites.

Speed of Light

At the core of special relativity is the speed of light, a constant that stands unchallenged as the cosmic speed limit. This immutable speed, roughly 299,792 kilometers per second, is not just another number but a key player in the laws that govern our universe.

The speed of light in a vacuum, often denoted by the symbol 'c', is the yardstick by which we measure enormous cosmic distances and it's the same for all observers, no matter how fast they're moving. Whether you're stationary or zooming through space, light beams will always pass you at speed 'c'. This constancy leads directly to phenomena such as time dilation and length contraction. It's the reason why the Lorentz factor depends on the ratio of an object's velocity to the speed of light.

In our textbook exercise, the motion of a clock at a fraction of the speed of light gives us a direct application of the speed of light's role in special relativity. It's not just about measuring distances; it's about understanding how movement at such high velocities affects the very fabric of time.