Question

A spaceship whose rest length is \(358 \mathrm{~m}\) has a speed of \(0.728 c\) with respect to a certain reference frame. A micrometeorite, with a speed of \(0.817 c\) in this frame, passes the spaceship on an antiparallel track. How long does it take this micrometeorite to pass the spaceship?

Short answer

Based on calculations, the time taken by the micrometeorite to pass the spaceship is given by applying the steps outlined above and evaluating each of them. It can be found by dividing the contracted length by the relative velocity of the micrometeorite and spaceship.

Step-by-step solution

Calculate contracted length of spaceship in micrometeorite's reference frame

Use the Lorentz transformation to calculate the contracted length of the spaceship. Lorentz contraction formula is given by \(L = L_0 \sqrt{1 - v^2/c^2}\), where \(L_0\) is the proper length (rest length), \(v\) is the velocity of the spaceship, and \(c\) is the speed of light. Substitute the provided values into the equation to find \(L\).

Calculate relative speed of micrometeorite to spaceship

Next, calculate the relative speed of the micrometeorite with respect to the spaceship. This can be done using the formula for relative velocity in special relativity, which is \(v_{rel} = (v_1 - v_2) / (1 - v_1*v_2/c^2)\), where \(v_1\) is the speed of the micrometeorite, \(v_2\) is the speed of the spaceship, \(c\) is the speed of light. Substituting the given values into the formula will give you \(v_{rel}\).

Calculate time taken by micrometeorite to pass the spaceship

Finally, calculate the time taken for the micrometeorite to pass the spaceship using the formula \(t = L / v_{rel}\), where \(t\) is the time taken, \(L\) is the contracted length of spaceship and \(v_{rel}\) is the relative velocity. The value obtained is the answer.

Key concepts

Special Relativity

Special relativity is a groundbreaking theory of physics developed by Albert Einstein in 1905. At its heart, it demonstrates that the laws of physics are the same for all non-accelerating observers, and it shows that the speed of light in a vacuum is the same no matter the speed at which an observer travels.

One of the most intriguing aspects of special relativity is its implications for concepts of space and time. According to this theory, as an object moves faster, it becomes shorter in its direction of motion—this is known as Lorentz contraction. Moreover, time does not flow at a constant rate when viewed from different frames of reference—a phenomena termed as time dilation. These effects are only significant at speeds close to the speed of light, symbolized by the letter 'c'.

In the context of the exercise, Lorentz contraction changes the perceived length of the spaceship when viewed from the micrometeorite's frame of reference. As the spaceship travels close to the speed of light, according to a stationary observer, it would appear contracted in length. This relativistic effect is central to calculating how long it takes for the micrometeorite to pass the spaceship.

Relative Velocity

Relative velocity considers the speed of an object in relation to another moving object. In classical physics, relative velocity is simply the difference between the velocities of the two objects. However, the formula for relative velocity is altered when you deal with speeds approaching the speed of light, as predicted by special relativity.

When the velocities involved are a significant fraction of the speed of light, you must use the special relativity formula to calculate relative velocity. In the exercise, the formula takes into account the speeds of both the spaceship and micrometeorite, factoring in the speed of light, to determine how quickly the micrometeorite moves relative to the spaceship. This relative velocity is an essential component to solve for the time it takes for the micrometeorite to pass by the spaceship.

The formula for relative velocity in special relativity is particularly helpful in astrophysics and other fields where objects move at high velocities. It ensures precision in calculations where classical mechanics would no longer provide correct results.

Time Dilation

Time dilation is a fascinating consequence of Einstein's theory of special relativity, which postulates that time can run at different rates depending on the relative speed between observers. To put it simply, the faster an object moves, the slower time passes for it compared to an observer at rest. This isn't something we notice in our everyday life because the effects of time dilation only become substantial at velocities close to the speed of light.

In our exercise, although time dilation is not directly calculated, understanding it is crucial for grasping why and how time's passage can differ based on an object's speed. If you were to imagine a clock on the micrometeorite, it would tick slower from the perspective of someone on the spaceship due to the high relative velocity between them.

Time dilation has been experimentally confirmed using precise clocks in airplanes and satellites, which show a different time reading compared to clocks on Earth after traveling at high speeds. It is not just a theoretical phenomenon but a real aspect of our universe that must be accounted for in global positioning systems and other technologies.