I've got a little question here. Basically we've got this guy named "B" driving a straight track with a high constant velocity u. His buddy, "C" on the side of the track measures B's fuel consumption to be dn/dt. Now C filled the tank with just enough gas to make it the distance of the track. What rate does "B" measure for fuel consumption? So what happens to B's car in the end. Does he make it, come up short or what? -Well here is my crack at it. I'm not exactly sure how dn/dt will look in B's frame. Maybe altered by the gamma factor. One thing for sure is that if B is hauling then obviously relativity takes over. C would see B travel gamma times further. So would C think he ran out of fuel? But in B's frame, the proper time interval.... ugh... I dunno. I suppose thats why I'm asking. Thanks in advance!
How are you measuring fuel consumption? It would seem to be to be difficult to measure the mass of the fuel on a moving train, so I'd guess maybe you're measuring the volume? Or are you conceptually counting molecules (i.e. wanting to know how many moles of fuel are left). If you just put the fuel in a vertical tank of constant cross-sectional area, and measure the height of the tank, you'll have a measure of the fuel which won't vary with motion, but I assume that's too simple a solution or not what you're looking for?
if you think about it. C is a stationary observer measuring fuel consumption for a moving frame of reference "B" going at speed u. "C" would measure fuel consumption as a lot more than it actually is, since the frame is moving and the guy "B" is having a slower time than outside. Making the fuel consumption less than observed by the outside.
I'm still confused about how fuel consumption is being measured - I'm assuming for the time being one is just looking at a "fuel gage" which is vertical to the direction of the motion, or looking directly at the height of the fuel in a transparent fuel tank. From the viewpoint of the stationary observer, the clock on the moving train would be ticking slowly. So the stationary observer sees the train as having a slow clock, but moving along a full-length track with no length contraction. So the fuel is flowing longer, but at a slower rate, than the case below. The moving observer thinks his clock is just fine, thank-you-very-much, but sees a track that's length contracted. The result is that everyone agrees on how much fuel is used at the end.
The rate is droplets of fuel injected in the carburetor per second. As to how the heck C measures the fuel consumption from where he standing, I dunno. But I dont think it matters. What matters is that he is gonna see a different distance which means his fuel assumptions arent gonna work out. Will C see a longer distance? Gamma times further...? therefore we would think the fuel would run out??? But physically the distance would be traversed so he would end up at the finish line in the end. ---> u (constant) ----------------------track--------------------- Pit Stop: C
Yes, C would see a onger distance traveled than B actually traveled, but C will also see a different rate of fuel consumption than B. He will see a slower rate than B. In each case, both observers are right when they say that the car DOES wind up finninshing the race.