Special Relativity - Is damage absolute?

[pbj_sweg]

1. The problem statement, all variables, and given/known data
A car of proper length 12m is being driven at 0.9c through a garage of proper length 6m. The garage has a front and back door. The garage owner, Joe, says that the car will fit inside the garage with no damage to it, albeit for a tiny amount of time. The car driver, Julie, says this is impossible, and the car will be damaged.

a) Show, with calculations, that you understand the basis of both the garage owner's and the driver's claims.
b) Event 1: The back of the car just clears the front (entrance) door of the garage.
Event 2: The back (exit) door, from the driver's perspective, is shut.
The driver yells that the exit door, when it closed to encapsulate the entire car, damaged the car a particular distance away from the front bumper. Using these events and Lorentz transformations, find that distance, and show that according to the garage owner, there will also be damage to the same feature of the car. Conclude that damage is absolute.

Homework Equations

[/B]
$$ L' = \frac{L}{\gamma} $$
$$ \Delta{x'} = \gamma(\Delta{x}-V\Delta{t}) $$
$$ \gamma = \frac{1}{\sqrt{1-\frac{V^2}{c^2}}} $$

The Attempt at a Solution

a) Just applied the length contraction formula to the car first to get the car's length relative to the garage owner. I got omega to be 2.29, and the car relative to the garage owner to be 5.23m. Applied the length contraction formula a second time to get the garage length relative to the driver to be 2.62m. This makes sense because the owner thinks the car will be able to fit while the driver thinks the garage is too small.

b) I don't know how exactly to use the transformation formulas now because based on part a, the garage owner seems to be able to fit the car inside the garage without any damage. I know that pictures are not preferred and not recommended, but I have no idea how to format the work I have written so I hope you guys understand.

Thank you so much!

 

[phinds]

Your figure is too tiny to be readable.

 

[pbj_sweg]

Your figure is too tiny to be readable.

Sorry about that, I've attached one more picture. If it's still unreadable, I'll type everything out. Thank you.

 

[PeroK]

I can't read your notes fully. One thing you may not have done is set up a coordinate system. To use the Lorentz Transformation you need to have a common origin (time and place).

This problem looks quite difficult if you haven't already mastered the simpler "garage door" problem.

 

[pbj_sweg]

I can't read your notes fully. One thing you may not have done is set up a coordinate system. To use the Lorentz Transformation you need to have a common origin (time and place).

This problem looks quite difficult if you haven't already mastered the simpler "garage door" problem.

My equations were the following:

- $\Delta{x} = L$ where L is the proper length of the garage
- $\Delta{t} = ?$ because I'm assuming that these events are simultaneous for Julie and therefore not simultaneous for Joe
- $\Delta{x'} = L'_{car} -L_d$ where $L'_{car}$ is the proper length of the car, $L_d$ is the length of the car sticking out of the garage for Julie, and $\Delta{x'}$ is the garage length relative to Julie
- $\Delta{t'} = 0$ because the events are simultaneous for Julie

I know that the proper length of the car is 2.62m from my previous calculations, so I just plugged that into my third equation to get $L_d$ to be 9.38m. Then, I used length contraction to get that 9.38m relative to Joe is 4.1m (omega is 2.29).

Having said all that, I'm almost 100% sure this is incorrect because as you said, I don't have a common origin.

Would it be sensible to say that event 1, the back of the car clearing the entrance of the garage, is simultaneous for both Joe and Julie?

 

[PeroK]

There are a lot of problems with your previous post.

First, a single event cannot be "simultaneous". Only two or more events can be simultaneous in a given frame of reference.

Second, length contraction and time dilation require you only to know the relative velocity between the reference frames. But, to deal with events at a defined time and place you need to specify a coordinate system.

That's why you can plug numbers into your formulas but can't go any further.

I would suggest you leave this problem and revise the Lorentz transformation and how to set up coordinate systems. Then come back to it.

Haven't you got some better problems just on the Lorentz Transformation?

This is quite a hard problem for now, I think.

 

[PeroK]

PS re Lorentz. This is a transformation of coordinates, not of lengths and time intervals. For example:

$t' = \gamma(t - \frac{Vx}{c^2})$