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Title: GR8677.069
Post by: conantee on January 06, 2010, 12:16:57 PM
Question 69-71
A car of rest length 5 meters passes through a garage of rest length 4 meters. Due to the relativistic Lorentz contraction, the car is only 3 meters long in the garage's rest frame. There are doors on both ends of the garage, which open automatically when the front of the car reaches them and close automatically when the rear passes them. The opening or closing of each door requires a negligible amount of time.

69. The velocity of the car in the garage's rest frame is
(A) 0.4 c
(B) 0.6 c
(C) 0.8 c
(D) greater than c
(E) not determinable from the data given

Title: Re: GR8677.069
Post by: conantee on October 13, 2010, 05:45:47 AM
First, you can eliminate (D) out since any object cannot exceed the speed of light. (E) "not determinable from the data given" is not a famous for correct answer. If you forget everything about relativity, just guess.

However, do a wise guess. The velocity of the car should be obtained from information about the car itself. The number involved with the car is the rest length 5 \mbox{ m} and the length measured in the garage's rest frame 3 \mbox{ m}. This phenomena is called "length contraction". The contraction rate governed by a factor related to speed of the car. This factor, if you are familiar with it a bit, is just not simply v/c. So, (B) is wrong.

Actually, that factor is \sqrt{1-v^2/c^2}. So, we need to solve
 \frac{3}{5} = \sqrt{1 - \frac{v^2}{c^2} or \frac{3^2}{5^2}+ \frac{v^2}{c^2} = 1

The answer is therefore \frac{4}{5}c = 0.8c

The correct answer is (C). 72 of 100 people answered this one correctly.