[A modern Zeno problem.] Achilles runs a total distance D=2^k meters in a total time T seconds. For example, Achilles might run 2048 meters in 491.52 seconds (= 8:11.52). Achilles wasn't terribly fast by today's world record standards. :-) Achilles's average *pace* (=1/speed) for the first half (=D/2) is 5 seconds/500m (= 0.01 seconds/meter) *faster* than his average pace for the second half. Achilles's average pace for the first quarter (=D/4) is 0.01 seconds/meter faster than his average pace for the second quarter. and so forth. Achilles's motion is a smooth analytic function of time and/or distance, excepting perhaps the origin. What is the functional form of his motion -- e.g., distance as a function of time, time as a function of distance, speed as a function of time, speed as a function of distance, pace as a function of time, pace as a function of distance -- any one of these forms should be useful. If I haven't made a mistake, Achilles's absolute *average pace* over the entire run shouldn't matter; clearly if the average pace for the first half differs by 5 seconds/500m from the average pace for the second half, then we could simply set the average pace for the entire run to zero (i.e., Newton relativity), so he may end up going backwards part of the time (i.e., relative to a tortoise moving at constant pace = Achilles's average pace). But this still leaves the problem of the functional form of his speed.