When logic and calculus collide

[GRaduns340]

It makes sense.

I'm not sure your assumption of the starting velocities being the same is completely true though. Technically, there is no derivative of the position graph at zero, because the derivative doesn't exist at the endpoints of a graph. Logically, yes, their velocities are both 0, but in order to use a derivative to prove it you would have to assume that the runners were both at rest before the start of the race, but with a lack of that knowledge there is no way to find f'(0).

You'd actually have to prove that it was an intermediate time at which their velocities were the same, like Tim suggested. I don't know if you've done it yet, but the mean value theorem would be useful there. (I think I pulled out the right name).

[Lil' Lavery]

Originally Posted by GRaduns340

It makes sense.

I'm not sure your assumption of the starting velocities being the same is completely true though. Technically, there is no derivative of the position graph at zero, because the derivative doesn't exist at the endpoints of a graph. Logically, yes, their velocities are both 0, but in order to use a derivative to prove it you would have to assume that the runners were both at rest before the start of the race, but with a lack of that knowledge there is no way to find f'(0).

You'd actually have to prove that it was an intermediate time at which their velocities were the same, like Tim suggested. I don't know if you've done it yet, but the mean value theorem would be useful there. (I think I pulled out the right name).

Yeah, we used the Mean Value Theorem to prove it via calculus. And according to calculus, their starting velocities don't have to be equal to 0, but I know calculus can prove it, I was wondering if it worked logically.
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.

[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.

[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.

[Kevin Sevcik]

Originally Posted by 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).

Ken, you've only shown that this theorem is only true for continuously differentiable functions of position. Your first argument is true if and only if the runner can go from 0 velocity to high velocity instantly, that is, without passing through any velocities in between, with infinite acceleration. That's physically impossible.

Starting time doesn't make a lick of difference. If one starts a second after the gun and runs at speed A, and the other starts 10 seconds late, he has to run at speed A+foo to get to the finish line at the same point. And you can't physically get from 0 to A+foo without going through A at some point.

Arguments about them not starting at the same time or place are beside the point. If you want to argue that the world's not perfect well fine. Toss all your fancy mathematics and physics in the trash because you can come up with some real world situations that are completely different from the stated problem.

[KenWittlief]

well, the problem was stated in terms of humans running a footrace, not in terms of imaginary dots moving along perfectly straight, perfectly parallel lines.

As Rich pointed out, the runners do not have to start from a stand still. In sailing races, and in Nascar the racers are moving when the race begins. As long as you dont cross the starting line before the opening shot your speed is up to you. If you are moving towards the starting line before the race starts, your speed is not zero, and it may not be zero at any point during the race (from 'Go' to the finish line).

Also, runners cant possibly stay exactly on the perfect centerline of a race course, so one person will end up running a longer distance than the other, therefore he could run faster for the entire race, and still finish with a tie.

This problem being worded this way brings up an interesting point: the math we learn in college is based on linear systems, but in the real world almost nothing is linear. So while we try to make the math work out to the nth degree of accuracy and precision (the right answer), in the real world the best we can do with math is approximate the non-linear physical world.