When logic and calculus collide

[KenWittlief]

starting velocity is easy to straighten out. They dont both have to start at the same time. One person might be late for the race, and run up to the starting line and run across it, so his starting velocity was not zero.

But that means he could run faster for the whole race and just catch up at the finish line. They never ran at the same speed at any point.

thinking they both started at the same time is a mistaken assumption. In fact, according to the laws of physics it would be impossible for them both to start at exactly the same instant (point in time). (action and reaction - time for sound to reach the ear of both runners - human response time for sound to cause legs to move... the two runners could never start perfectly synchronised, one will always start before the other).

This kills the proof already.

The second assumption is that both runners ran exactly the same distance. For this to be true one would have to run exactly behind the other, in exactlyt the same path, or they would have to run in a straight line. If one runner wanders off the path of the course the slightest bit, he has to run faster for the race to be a tie, so again, he could run faster for the whole race, and still end in a tie

a third assumption: that to speed up and slow down you must cross through the intermediate speeds - that you ramp up, and ramp down. True when speeding up, but if you run into a lamp post, brick wall, or parked car, your velocity instantly goes to zero. The graph of your velocity would have a discontinuity at that point, and would therefore be undefined (accelerated acceleration, or jerk). At that instant your velocity instantaniously goes from, lets say 12 mph, to 0.

There is no logic to the proof. In theory: yes. In reality: no.