When logic and calculus collide

[Greg Marra]

Originally Posted by Lil' Lavery

I suppose that the two speeds would have to be equal at some point, but I can't help this feeling that there's some way that it could be done with never having the same speed.

Picture a plot that contains both runners velocities plotted against time.

Velocity is (m/s), right? So when we integrate that against time (the x-axis) we get just (m). This means that the area under the velocity curves for the two runners is the distance they have run. They run an equal distance, so the area under the two curves has to be the same.

Draw two curves such that the area under the two are the same, that one racer finishes before the other, and that they never cross. If the curves cross, it means they have the same velocity.