The car-in-the-garage problem, car man has just purchased the world's longest stretch limo, which has a proper length of Lc = 30.5 m. In Figure a, 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, Garage man, 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 Garage man??s scheme, an xc axis is attached to the limo, with xc = 0 at the rear bumper, and an xs 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 x, reference frame; Garage man 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 Garage man: tgl: tcl = 0. The event occurs at xc = xs = 0. Figure b shows event 1 according to the xg reference frame. Event 2: When the front bumper reaches the back door, that door opens. Figure c shows event 2 according to the x, reference frame. According to Garage man,

(a) What is the length of the limo, and what are the space-time 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 x, 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 is the space-time 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?

(l) Finally, who wins the bet?

Transcribed Image Text:

(a) (b) (e)