Two racers start in between two finish lines of equal distance 'd' , ie a distance 2d sits between the two lines with the racers sitting a distance d from each. Refferees who time the race in the earth frame are able to say that both racers start at time=0s and at the same position of x=0. Both racers run a distance d in opposite direction with constant speed 'u' and in the refs' frame both racers finish at the same time.
Using Lorentz transforms determine the finishing time of each racer in the racer who is traveling left (call him racer one)'s reference frame.
Lorentz transform equations:
The Attempt at a Solution
So if we are attempting to find the (t final)' in the primed reference frame, the frame that is moving at a speed of -u I first thought to find the time that it took for each racer to finish in the unprimed frame (ref frame) and then transform it to the primed frame.
so in the unprimed frame it should take each racer t=d/v=d/u time to finish the race.
In order to use the lorentz transform correctly to get to the frame where the speed of the new frame is -u and racer one is travelling at speed=0 I thought to transform racer two's velocity into his velocity in the primed frame.
so then t final for this racer would be the lorentz time transformation as listed above useing this value of u(racer 2)' for gamma.
timefinal(racerone) would have gamma equal to one since he is not moving in this frame so
meaning that the difference is times for finishing the race in the first racers frame is equal to the time racerone thought he finish multiplied by the calculated value of gamma above.
I was told that this is incorrect, but cannot seem to find my mistake.
I don't know how comfortable I am saying that gamma is equal to one for racerone, but since the velocity of this racer is 0 in this new frame why is this not correct?
I'm at a loss,
Thanks for any and all help