Can a Limo Fit into a Smaller Garage with Special Relativity?

[bcjochim07]

Homework Statement

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?

Homework Equations

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

Event A
t' = 0

Event B
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.

 

[bcjochim07]

Any thoughts?

 

[thomate1]

The Special Relativity Paradox arises when we try to apply the principles of special relativity to scenarios that seem to defy common sense. In this case, Carmen and Garageman's argument about fitting the limo into the garage may seem contradictory, but it can be resolved by understanding the concept of length contraction and time dilation in special relativity.

In the rest frame of the limo, the garage appears to be shorter due to length contraction. This means that the limo can fit into the garage without any issues. However, in the rest frame of the garage, the limo appears to be longer due to length contraction. This is where the paradox arises - how can the limo fit into the garage if it appears to be longer?

The key to resolving this paradox lies in understanding that different observers in different frames of reference will measure different lengths and times. In this scenario, the garage and the limo are in different frames of reference, so their measurements of length and time will differ. This is because the speed of light is constant for all observers, and as a result, space and time must adjust accordingly to maintain this constant speed.

To find the speed of the limo with respect to the garage, we can use the Lorentz transformations and the given proper lengths to calculate the speed in the garage's frame. As you have correctly calculated, the speed required for the limo to fit into the garage is 0.9978c.

In conclusion, the Special Relativity Paradox can be resolved by understanding that different frames of reference will measure different lengths and times due to the principles of special relativity. This allows for seemingly contradictory scenarios, such as the limo fitting into the garage, to be possible.