Carmen has just purchased the world's longest stretch lim, which has proper length 30 m. A garage has a proper length of 6.0 m. Carmen concludes that there is no way to fit the limo into the garage. Her buddy Garageman claims that under the right circumstances the limo can fit into the garage with room to spare, all you have to do is speed the limo up until the moving limo takes up one third of the proper length of the garage. The front garage door closes just behind the speeding limo, and the back garage door opens just in front of the speeding limo.
1) Find the speed of the limo with respect to the garage required for this scenario.
2) Carmen protests that in the rest frame of the limo, it is the garage that is Lorentz contrated. As a result, there is no possibility whatsoever that the limo can fit into the garage. What could be the possible basis for resolving this paradox?
The Attempt at a Solution
1) 2.0 m = sqrt(1-B^2)*proper length
B = .9978 c
For part 2, I am struggling to figure out which is the proper frame.
I used the Lorentz transforms for simultaneous events happening at t=0)
Event A (back of car right next to front door)
x' = 0
Event B (front of car at L/3) where L is length of garage
x' (L/3)/ sqrt(1-.9978^2) = 5.028 L
t' = 0
t' = (0-(.9978)(L/3)*(1/c^2))/sqrt(1-.9978^2) = -5.017 L/ c^2
I'm confused about what these numbers tell me. Could someone give me a push in the right direction? Thanks.