Originally Posted by sloth
Two cars leave point A traveling in the same direction at 2mph. Relative to car 1, car 2 is motionless.
Two cars leave point A traveling in the same direction at C. Relative to car 1, car 2 is... what?
Please explain this in normal dumb guy words. I am not a rocket surgeon, nor am I impressed by them.
There is no way to explain in "normal dumb guy words". It is the special theory of relativity. It's easy and graspable for people willing to work for it. "Please explain this in normal dumb guy words" means "I am not willing to work for it". If you are willing to work, then here it goes.
You are using the principle of relativity in your analysis. This goes back to Galileo. Say that at time t_0, both cars are at x_0. Then at time t, both cars are at x=t*2mph if we measure t in hours.
That is the "restframe" of point A. The position of A is always x=0 when measured in this rest frame.
The principle of relativity essentially says that physics works the same in all reference frames which are the rest frame of a particle moving with constant velocity. We need to be able to translate between frames.
Lets move to the car frame. With the cars moving at 2mph to the right in A's rest frame, we see that in the car frame, the cars are always at x=0. On the other hand, we know from experience that the distance between them will stay the same from one frame to the other. Hence the position of of A in the car frame must be given by x = -t*2mph. So in the car frame A is moving to the left at 2mph.
In general, if (t', x') is the coordinate of an event in A's frame, and (t, x) is the coordinate of an event in the car frame, then t = t' and x = x - t*2mph. These are the galilean transformations.
The problem occurs because the speed of light is predicted to be constant from maxwell's equations and is experimentally confirmed to be constant.
So c is constant.
But if those transformations are correct, then we're in trouble. Flash a light at (0, 0). Describe it in A's frame. A photon at (0, 0) will also intersect the event (1, c), one hour later and one light-hour away. Now lets translate to the car frame. A photon at (0, 0) and (1, c) in A's frame will be present at (0, 0) and (1, c - 2) in the car frame. So the car frame will measure the speed to be c - 2 miles per hour. This contradicts experiment and the predictions of electrodynamics.
The correct solution is to use the lorentz transformations instead of the galilean ones. This has the effect of preserving the principle of relativity but sacrifices some common misconceptions of distance and time. The key to accepting this is to understand that you don't know everything and that nobody ever made a deal with you promising that your experience was valid for all realms of existence.
If you think hard and ask questions instead of making ignorant statements, we can go into the lorentz transformations at some later date and you might actually learn something...
EDIT:
Of course the erroneous assumption was "the distance is the same in all frames of reference". Also, the assumption that t = t' is erroneous as well. There is no "absolute" time.
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