Physics Formula for Force

My friends have been a little old for college. But it has been a second chance for them, or rather a third one...

So this time they wanted to do it right.

Unfortunately, their curriculum included Physics. Colleges usually discourage specialization. Colleges want graduates to be more or less universally educated.

They were my friends, so quite naturally they turned to me to get the extra explanations they felt they'd need to swallow that indigestible stuff. Teach me physics. And I explained. And waved my hands. And ... - and I finally realized that it was not physics in the first place, but some concepts of Maths they had not taken on! Especially the concepts of derivation and integration. My gosh!

So I went there. I started to talk about derivatives. And that a derivative basically is a slope. And that you can visualize a derivative as the slope of the tangent to the point in question.

Did I mention that my friends were slightly above normal college age? If you are slightly above college age, you have learned that only things matter which have a bearing on your very life. Subconsciously, that is. Consciously all you realize is that you feel very uncomfortable when you try to follow a logical sequence of thoughts without seeing the benefit. You find it VERY hard to do exercise problems, because they are good for nothing (except exercise), the solution is already known, and the situation is generally so oversimplified that this well-known solution does not correspond to any real life solution of any real life problem.

So they came back on me: "But WHAT is a derivative GOOD for in the end?"

Well, I understood. I had to find an example of their daily struggles to motivate them to get acquainted with the idea of infinitesimal calculus.

So I went on: "Just imagine: You drive your car from point A to point B. let's say the distance is 20 miles. And you take 20 min to arrive. How fast are you?" They got the answer immediately: "60 Mph !". But I was not content. "Really? Look, what happens - you start your car. Do you instantly jump to 60 Mph? And after arrival, you normally don't hit a brick wall and come to an instant halt, do you?"

They got the idea. "No, of course not, we have to start and first pick up speed, and than before we arrive, we have to slow down and gradually come to a stop. And if we have to do all in twenty minutes, our speedometer will be above 60 somewhere in the middle of the journey."

That was the word for which I had hoped: Speedometer!

"So you say you drive above 60 miles per hour, but not for an hour, you are actually only moving a couple of minutes! And even then, your speed is not constant, bur rather varies constantly!!"

"What are you driving at with all that imaginary driving??"

"Well, didn't you ask what a derivative is good for? Here you have it: The speedometer shows you the derivative of your local displacement with respect to time, because that's what speed is. And if you look at the needle of the speedometer and how it moves when showing your variable speed, the speed of that needle movement goes with the second derivative, the acceleration!"

Now we got it. Finally a sigh: "So there is a derivative built in to my car!"