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Chapter 35: Problem 14

At rest, a rocket has an overall length of \(L .\) A garage at rest (built for the rocket by the lowest bidder) is only \(L / 2\) in length. Luckily, the garage has both a front door and a back door, so that when the rocket flies at a speed of \(v=0.866 c\), the rocket fits entirely into the garage. However, according to the rocket pilot, the rocket has length \(L\) and the garage has length \(L / 4\). How does the rocket pilot observe that the rocket does not fit into the garage?

Short Answer

Expert verified
Explain your answer. Answer: No, the rocket does not fit into the garage from the rocket pilot's perspective. This is because, from the pilot's point of view, the garage is contracted to a length of L/4 due to length contraction in special relativity, while the rocket is still L in length. Since the contracted length of the garage is shorter than the length of the rocket, the rocket cannot fit into the garage from the pilot's perspective.

Step by step solution

01

Calculate the Lorentz factor

First, we need to find the Lorentz factor, which is given by the formula \[ \gamma = \dfrac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \] Plug in the given value of \(v = 0.866c\), we obtain, \[ \gamma = \dfrac{1}{\sqrt{1 - \frac{ (0.866c)^2}{c^2}}} = 2\]
02

Calculate the contracted length of the rocket from the rocket pilot's perspective

According to the problem, the rocket pilot observes the rocket's length to be \(L\), which means there is no length contraction of the rocket from the pilot's perspective.
03

Calculate the contracted length of the garage from the pilot's perspective

Length contraction formula is given by \[ L' = \dfrac{L}{\gamma} \] where \(L'\) is the contracted length, \(L\) is the original length, and \(\gamma\) is the Lorentz factor. As the garage has a length of \(L/2\) when it is at rest, the rocket pilot observes the garage to have a length of \[ L' = \dfrac{L/2}{\gamma} = \dfrac{L/2}{2} = \dfrac{L}{4}\]
04

Determine whether the rocket fits into the garage

From the rocket pilot's perspective, the garage has a length of \(L/4\), while the rocket has a length of \(L\). Thus, the rocket does not fit into the garage from the pilot's point of view.
05

Conclusion

In this exercise, we have used the concept of length contraction in special relativity to examine the situation where the rocket pilot perceives their rocket to not fit into the garage. By calculating the Lorentz factor and the contracted lengths of the rocket and garage, we've shown that from the rocket pilot's perspective, the rocket doesn't fit into the garage because the garage appears to be only \(L/4\) while the rocket is still \(L\).

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Most popular questions from this chapter

In the twin paradox example, Alice boards a spaceship that flies to a space station 3.25 light-years away and then returns with a speed of \(0.65 c\). a) Calculate the total distance Alice traveled during the trip, as measured by Alice. b) With the aforementioned total distance, calculate the total time duration for the trip, as measured by Alice.

Two twins, \(A\) and \(B\), are in deep space on similar rockets traveling in opposite directions with a relative speed of \(c / 4\). After a while, twin A turns around and travels back toward twin \(\mathrm{B}\) again, so that their relative speed is \(c / 4\). When they meet again, is one twin younger, and if so which twin is younger? a) Twin A is younger. d) Each twin thinks b) Twin \(B\) is younger. the other is younger. c) The twins are the same age.

Suppose you are explaining the theory of relativity to a friend, and you have told him that nothing can go faster than \(300,000 \mathrm{~km} / \mathrm{s}\). He says that is obviously false: Suppose a spaceship traveling past you at \(200,000 \mathrm{~km} / \mathrm{s}\), which is perfectly possible according to what you are saying, fires a torpedo straight ahead whose speed is \(200,000 \mathrm{~km} / \mathrm{s}\) relative to the spaceship, which is also perfectly possible; then, he says, the torpedo's speed is \(400,000 \mathrm{~km} / \mathrm{s}\). How would you answer him?

In your physics class you have just learned about the relativistic frequency shift, and you decide to amaze your friends at a party. You tell them that once you drove through a stop light and that when you were pulled over you did not get ticketed because you explained to the police officer that the relativistic Doppler shift made the red light of wavelength \(650 \mathrm{nm}\) appear green to you, with a wavelength of \(520 \mathrm{nm}\). If your story had been true, how fast would you have been traveling?

At what speed will the length of a meter stick look \(90.0 \mathrm{~cm} ?\)
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