A horse starts running on an infinite road with a bell tied to his tail. His initial speed is 1m/sec and he runs at 1-meter jumps. After every jump, the bell rings and once he hears the ringing he doubles his speed. Assuming that the horse can run infinitely fast, find the final speed after running a distance of 400 kilometres.
I assume the problem is related to the speed of sound. After each second, the sound waves travel behind the horse and reach him almost instantly at the beginning, but after his speed is >340 m/sec (speed of sound) the waves of sound are travelling behind him but never reach him, since the horse is faster. After the first meter, the bell rings and travels 1m in 1/340 sec, but during this time (1/340 sec) the horse has advanced 1/340 m, so the horse will hear it in 341/340 seconds after he started running. Then at this point he doubles his speed. Any ideas about the final equation? We also know that he runs at 1-meter jumps so if he hears the bell after 1/340 sec, it does not matter, as this will affect his motion only at the next jump, right?
The speed keeps doubling as long as the speed is below 340 m/s, so that would be 512 m/s ... and 400km is way more than enough length for the first 8 bell rings to catch up with the horse. Even when the horse runs at 256 m/s, and assuming the distance between the bell and ears is, say, 3 meters, then it will only take a fraction of a second for the sounds to reach the horse's ears, and hence the horse will have traveled less than 256 meters before hearing the sound .. so yeah, 400km is plenty to reach 512 m/s