When will Peter catch up the light beam?

Question

Peter run at 400,000km/s. Light beam runs at 299,792km/s. If Peter is 100000km behind the light beam, when will Peter catch up the light beam?

Answer 1

Never.  According to relativity, no matter how fast Peter runs, he'll always see the light beam speeding away from him at the speed of light.  Also, you've got peter running faster than the speed of light, which again, according to relativity, isn't possible.

Answer 2

Peter the tachyon man running faster than the speed of light will catch up to the light beam in about 1 second to an observor in the rest frame at which Peters velocity is measured (0.997924 seconds to the 6 significant figure accuracy that you have given in the question). If Peter is losing energy to cherenkov radiation then his speed will increase to infinity and he'll probably get there a bit quicker, but I'll assume he's going at a constant velocity.

In Peter's frame of reference time is a little bit more complicated. Being a tachyon man he travels in imaginary time as defined by the lorentz equations (or he may be massless in which case he is not bounded by relativity).

In his frame it will take 0.997924/sqrt(1-(400000/299792)^2) s

= +/- 1.12975i seconds

The plus or minus solutions are both equally posible and it will depend if he has negative energy travelling forward in time or positive energy travelling backwards in time.

What he will do when he catches up to it is uncertain as we don't know whether there can be any interaction between Peter and the world confined to slower than light velocities, or for that matter whether Peter actually exists, but if he does it'd take him +/- 1.12975i seconds