Solved: The car-in-the-garage problem. Carman has just

Chapter 37, Problem 69

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The car-in-the-garage problem. Carman has just purchased the world's longest stretch limo, which has a proper length of Lc = 30.5 m. In Fig. 37-32a, it is shown parked in front of a garage with a proper length of Lg = 6.00 m. The garage has a front door (shown open) and a back door (shown closed). The limo is obviously longer than the garage. Still, Garageman, who owns the garage and knows something about relativistic length contraction, makes a bet with Carman that the limo can fit in the garage with both doors closed. Carman, who dropped his physics course before reaching special relativity, says such a thing, even in principle, is impossible. To analyze Garageman's scheme, an Xc axis is attached to the limo, with Xc = 0 at the rear bumper, and an Xg axis is attached to the garage, with Xg = 0 at the (now open) front door. Then Carman is to drive the limo directly toward the front door at a velocity of 0.9980c (which is, of course, both technically and financially impossible). Carman is stationary in the Xc reference frame; Garageman is stationary in the Xg reference frame.There are two events to consider. Event 1: When the rear bumper clears the front door, the front door is closed. Let the time of this event be zero to both Carman and Garageman: tgl = tel = O. The event occurs at Xc = Xg = O. Figure 37-32b shows event 1 according to the Xg reference frame. Event 2: When the front bumper reaches the back door, that door opens. Figure 37-32c shows event 2 according to the Xg reference frame.According to Garageman, (a) what is the length of the limo, and what are the spacetime coordinates (b) Xg2 and (c) tg2 of event 2? (d) For how long is the limo temporarily "trapped" inside the garage with both doors shut? Now consider the situation from the Xc reference frame, in which the garage comes racing past the limo at a velocity of -0.9980c. According to Carman, (e) what is the length of the passing garage, what are the spacetime coordinates (f) Xc2 and (g) tc2 of event 2, (h) is the limo ever in the garage with both doors shut, and (i) which event occurs first? (j) Sketch events 1 and 2 as seen by Carman. (k) Are the events causally related; that is, does one of them cause the other? (1) Finally, who wins the bet?

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