The problem sounds reasonable, but I seem to have shown there is no solution. At least if speed is a continuous function of position. Suppose we are on the real line R, headed toward 0, and the velocity at every x <> 0 is given by v(x) <> 0. Then for all x = u we require that the time it will take to reach 0 be 1 unit of time. Then Integral_{u to 0} of (1/v(x)) dx == 1, since that integral gives the time from x to 0. Let F(x) satisfy F'(x) == 1/v(x). Then F(0) - F(u) == 1 (for all u). Hence the function F(x) == const, so by the definition of F we have 0 == F'(x) == 1/v(x) and so v(x) == 1/0. Oops. --------------------------------------------------------- But come to think of it this derivation is not necessary. For if we begin at one hour away from our destination, then any length of traveling will take some positive time, and we must then be that amount *less* than one hour away. Contradiction. --Dan Simon wrote: << I was going back from a place about 100 km away from home and at one point the GPS I have was kind of stuck on a fix time of arrival, which later was corrected I presume. But it made me think of a problem which is the following. Suppose you are at exactly 100 Km from a place going on a straight road at a speed of 100 Km/hour. Obviously, it will take you 1 hour exactly to reach your destination. BUT : what would be your increase in speed or decrease IF , let's say you make a point every kilometer you made so far so that : It would be always 1 hour in time to reach your destination ? In other words, what is the change in speed IF at every kilometer there is always 1 hour left of driving until you reach destination. After 1 min at 100 km/hour, there is 99 kilometers left so if you need to reach destination in 1 hour you need to go at 99 km/hour. Is this a trivial problem ? My impression is that the speed would necessarly decrease to near 0 when you are 1 mm away from your home. In plain english , what is the function of the speed ? If someone has a solution, I would be glad to ear it.
________________________________________________________________________________________ It goes without saying that .