When logic and calculus collide

[Richard Wallace]

Originally Posted by Lil' Lavery

f'(t) was the difference of the two derivatives, not the derivative of either of the runners positons, and so could (and would have to be) zero. g'(t) and h'(t) were the velocities of the runners, and niether could have been zero (unless it was a race with absolutely no distance or where niether runner moved). The problem did state that both runners started at the same time, but you're right in that it may not mean they both had a velocity of 0 at the beginning.
I get the Calculus behind it (including the mean value theorem, which was the lesson where we were given this problem), I was trying to justify it logically, and I guess that it must be logically true. I've thought about it a little more and at one point they must both have been travelling the same speed.

Sean, I was pretty sure that you understood the calculus assignment, and the MVT. And I'm pretty sure you will be going much further into engineering math during the next few years. That's why I thought it worthwhile to quibble.

My quibble was with the assumption of any knowledge concerning f'(t) or g'(t) or h'(t) before the start and after the finish. Part of the quibble is practical: nothing about these derivatives is given in the problem statement. The other part is mathematical: Rolle's Theorem does not depend on any particular constraints on f'(t) at the endpoints of the interval.

I only jumped in here because I've seen this kind of error in setting up engineering math problems a few times before. The assumption that I'm quibbling over makes no difference to the proof sought in this problem, but there are many engineering problems for which incorrect assumptions about starting and/or ending conditions will lead to the wrong conclusion. And you are going to see them.

When I am a doddering old retiree and you are overseeing the design of infrastructure for the Martian colony, I want your mathematical reasoning to be perfect.

[DonRotolo]

Originally Posted by Richard

When I am a doddering old retiree and you are overseeing the design of infrastructure for the Martian colony, I want your mathematical reasoning to be perfect.

Yes, especially if that's where I'll be living...

Don

[Kaizer007]

I'm sorry this is four days late, but I wanted to ask how the last part of Ken's argument way up there makes sense:

"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."

I can't believe, (with only Physics and AP Physics class behind me *both High School Level*) that an instant stop is possible for running into a lamp post. I am to believe that even the slightest part of your body would start slowing down at a tremendous negative acceleration but not one that could be considered undefined. I had thought (from other math classes) that undefined acceleration could not be reached in this physical world.

[KenWittlief]

Originally Posted by Kaizer007

I'm sorry this is four days late, but I wanted to ask how the last part of Ken's argument way up there makes sense:

which part of the runners body would you measure to use as their 'speed' when they hit a brick wall? What system would you use to measure the speed from the instant that, lets say, the skin on their forehead first contacts the wall, and the point when their head comes to a compete stop?

Radar? GPS? a high speed camera? If the impact has a 1µS duration I dont think it can be measured by any means associated with running a foot race. For all practical purposes the acceleration is off the scale (infinite), compared to what a runner normally experiences.

It seems to be a mute point anyway. As others have pointed out, the two runners started 'at the same time', even though the laws of physics say this is impossible

and everyone appears to be assuming the two runners run exactly the same distance, even though this is also impossible.

The 'math class' answer is that they must run at the same speed at some point, including the starting velocity of zero. (because they are really two points moving along straight lines).

The physics (and logical) answer is that they dont have to run the same speed at any instant to end up in a tie.

In fact, as far as I know, there is no rule against a runner approaching the finish line, looping around and going back to the starting line, and looping back to the finish line, to run 3 times the distance as the other runner. This means he could run 18 mph for the entire duration of the race, while the other runner runs at 6mph for the whole race, and they both cross the finish line at the same instant.

This is why I am an engineer and not a mathematician. When you think outside the box on a math test you get an F !

[Kaizer007]

I'm sorry to have misrepresented my skepticism on the subject. I was not trying to make any point or mute point. My only inquiry was about acceleration.
You did point out that the negative acceleration would be off the charts when running into a lamp post and it would be much higher than a normal runner would be used to. Also it could not be measured by means in a foot race.
My objection is that infinity could not be reached negatively or positively in the real world.
A bullet, for example, stops instantly to our eyes, but it starts instantaneously to our eyes as well, and your argument stated that the ramp up is true, but not the ramp down. Even if it could not be measured, wouldn't logic state it couldn't get to be an infinite instantaneous stop if the runner hits a lamp post and just a very negative slope?

[KenWittlief]

Originally Posted by Kaizer007

...your argument stated that the ramp up is true, but not the ramp down. Even if it could not be measured, wouldn't logic state it couldn't get to be an infinite instantaneous stop if the runner hits a lamp post and just a very negative slope?

you are correct, in the normal sense of physics and math, even when you run into a brick wall you do not experience infinite acceleration to zero velocity. therefore as your velocity dropped to zero you would cross a point where your velocity would match the other runner (instantaniously).

I was only trying to show that when you try to apply pure mathematics to the physical world you almost always run into problems.

The math courses I took in college, when we were talking about things that I could correlate into physical realities, I could grasp them easily. Then when things became abstract and imaginary (6 dimensional space for example), I had a difficult time.

It was not until years after I graduated that I understood that most math we learn in college does not really apply to the real world (as I mentioned in a previous post).

To correct my example of an exception, an infinite force could stop a runner instantainously (infinite acceleration to zero velocity), in which case there would be a discontinuty in the velocity graph. Infinity is a valid number in math, but not in physics.

"Two runners start a race at the same time". Ok in math class, impossible in physics. The given statement is illogical in the real world. The question supposes an impossible situation, but they still want to know the 'correct' answer.

When Logic and Calculus Collide. The professors question itself is illogical.

[Kaizer007]

I absolutely agree and understand. Thank you.

[Michael Hill]

I don't know why I'm bumping this or was even reading this thread, but I can say that this is entirely possible. Just in higher dimensions. Nothing said they have to move in a straight path.