Question

A spaceship of rest length \(130 \mathrm{~m}\) drifts past a timing station at a speed of \(0.740 c .(a)\) What is the length of the spaceship as measured by the timing station? \((b)\) What time interval between the passage of the front and back end of the ship will the station monitor record?

Short answer

The length of the spaceship as measured by the timing station is approximately \(76.32 \mathrm{~m}\) and the timing station will record a time interval of around \(299.12 \mu s\) for the passage of the spaceship.

Step-by-step solution

Calculate the contracted length

To calculate the length of the spaceship as it drifts past the timing station, the length contraction formula can be used. This means that the contracted length \(L\) can be calculated as \(L = L_0 \sqrt{1 - v^2/c^2}\) where \(L_0\) is the rest length of the spaceship, \(v\) is the speed of the spaceship and \(c\) is the speed of light. Substituting the given values of \(L_0 = 130 \mathrm{~m}\), \(v = 0.740c\), and \(c = 1\) (because \(v\) is given as a fraction of \(c\)), we have \(L = 130 \mathrm{~m} \sqrt{1 - (0.740)^2} = 76.32 \mathrm{~m}\)

Calculate the time interval

To calculate the time interval between the passage of the front and back end of the spaceship, we need to use time dilation formula which states that \(\Delta t = \Delta t_0/\sqrt{1 -v^2/c^2}\) where \(\Delta t_0\) is the proper time, \(v\) is the speed of the spaceship and \(c\) is the speed of light. But first, we need to find proper time \(\Delta t_0\) which is the time interval recorded by an observer in the spaceship's frame. Assuming the spaceship is moving with constant speed, the proper time can be calculated as \(\Delta t_0 = L_0/v\) where \(L_0\) is the rest length and \(v\) is the speed. Substituting the values of \(L_0= 130\mathrm{~m}\), \(v=0.740c\), we have \(\Delta t_0=130\mathrm{~m}/0.740c = 175.68 \mu s\). Now substituting the values of \(\Delta t_0= 175.68 \mu s\), \(v=0.740c\) and \(c=1\) in time dilation formula, we have \(\Delta t =175.68 \mu s/\sqrt{1 - (0.740)^2}= 299.12 \mu s \)

Key concepts

Special Relativity

At the heart of modern physics lies the fascinating theory of special relativity, introduced by Albert Einstein in 1905. This revolutionary concept changed our understanding of space, time, and energy. Special relativity is based on two key principles: the laws of physics are the same for all observers in uniform motion relative to one another, and the speed of light in a vacuum is the same for all observers, regardless of their relative motion or the motion of the light source.

Within this framework, we encounter phenomena such as length contraction and time dilation, which are not perceivable at everyday speeds. As objects move closer to the speed of light, remarkable changes occur to their perceived length and the passage of time relative to stationary observers. These effects are crucial for understanding the behavior of particles in accelerators, the decay of cosmic rays, and even the workings of GPS systems, which must account for these relativistic alterations to ensure accuracy.

Through the lens of special relativity, time and space become intertwined, leading to the concept of spacetime. Events that appear simultaneous to one observer may not be so to another moving at a different velocity. This interdependence of space and time underscores the theory's revolutionary impact on physics.

Time Dilation

Time dilation is a direct consequence of special relativity and it describes how the perception of time can vary for different observers, particularly when they are moving at substantial fractions of the speed of light. According to this phenomenon, time as measured by a clock that is moving relative to an observer will be observed to tick slower compared to a clock that is at rest with respect to that observer.

This has profound implications for our understanding of time itself. It's not just a sequence of events unfolding at a fixed rate, but something that can stretch and compress depending on the relative speeds involved. Time dilation is often illustrated by thought experiments such as the 'twin paradox', in which one twin travels at high speed in a spaceship whilst the other stays on Earth. Upon the spacefaring twin's return, they would be younger due to experiencing less time passing - a real effect that has been experimentally verified using precise atomic clocks on airplanes and satellites.

For the exercise on the spaceship's passage past a timing station, time dilation explains why the time interval between the front and back ends of the ship would appear longer to station observers than to those on the moving ship. This can lead to the surprising realization that in the world of high-speed travel, time is not a universal constant but rather a variable that depends on your frame of reference.

Lorentz Transformation

To mathematically describe the phenomena of time dilation and length contraction, the Lorentz transformation equations come into play. These are set of equations that relate the space and time coordinates of one event as measured in two different inertial frames of reference that are in constant relative motion. Named after the Dutch physicist Hendrik Lorentz, these transformations are pivotal for translating measurements from one frame to another in accordance with the principles of special relativity.

The Lorentz transformation allows us to quantify exactly how observed lengths contract and time intervals expand depending on the relative velocity between observers. The usage of these equations is not just theoretical; they have practical applications in determining how fast-moving objects such as particles in a collider or spacecraft, like in our textbook exercise, behave. They ensure that despite the counterintuitive implications of special relativity, scientists can make accurate predictions and calculations that match experimental results.

The transformation involves factors of the speed of light and the velocity of the object, reinforcing the idea that as we approach light speed, time and space become increasingly relativistic. Understanding and applying the Lorentz transformation allows us to delve into deeply non-intuitive realms of physics where space and time are no longer rigid and absolute, but fluid and dependent on perspective.