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Now let me ...
Let me ...
Jump ... jump on aside just ...
... asking you a question.
And it's ... has no relating to the curve in 3-space. It's about the curve in a plane.
I will tell you about the relevance later.
If you are to build a road.
Let's say a train track.
And you want to do is to make a turn.
So you want a train to go straight here.
And then you want a train to go to there.
And then your task is to provide a curve here going from this point to that point.
So that a train can follow that curve and turn.
And the question is at the level of a curve.
Don't think about the two tracks ...
... and three dimentional nature of the things, right?
Just a curve.
What can you think about the curve?
What kind of curve would you suggest?
Student: [...].
Like ... for example?
Because you probably won't want the turn to happen like this, right?
So how would you like it to happen?
What would be an example?
Student: A quarter of the circle.
A quarter of the circle.
So instead of that corner that we're not willing to follow.
We want to follow a quarter of the circle.
Do you want it to be like this?
Well you said 'quarter' meaning that it touches.
It touches this line, it touches that line.
Is that going to be a good turn?
Assuming you can build the tracks perfectly straight.
And then perfectly circle-wise.
And then perfectly straight again.
Do you expect the passengers to enjoy the ride?
Do you expect the tracks to survive long?
For years?
Do you expect the train to be safe and not to break?
So those very questions are the main concern. Right?
So I have to tell you well there is a big problem there.
About this quarter of the circle.
Student: [...].
Well, the problem is not in the radius of the circle.
Well, thinl of the radius beeing ...
... a quarter of mile.
So it's a slow turn.
Not extremely slow. You still [look at] turn in reasonable time, right?
But ...
But the curvature is OK.
It's managable.
So a regular passenger should experience ...
... sideway acceleration on the way.
But that's OK.
The tracks should survive that acceleration.
The train should survive that force.
Everything is good with respect to the curvature there.
But the problem is not there.
Student: [...].
Exactly.
The problem is right here and right there.
So think about the passenger.
What that passenger would experience passing this point?
No acceleration on the side, right?
And then at this point suddenly bum!
Well the acceleration you experience there is OK, right?
When you push someone a little bit that's OK.
But when you suddenly hit, right? That's not OK.
And that's what the passenger will feel.
What will the tracks experience at this point?
Well this part of the track will experience no force left or right.
And that part of the track will experience ...
... that force, right?
The train beyond that point will push the tracks to the left.
So should it be expected the tracks to crack there?
Break?
You have to imaging massive train, right?
Even slowly moving.
That mass produces a lot of force.
And of course the train itself, right?
So the wheels experiencing no force on the side.
And then suddenly bum!
Such a hit on a wheel.
So from all points of view this point is not good at all.
So what was the key observation?
What is it exactly that made it bad?
It is not the force.
It is the observation that the force depends on time as you move.
And the observation was that the change of that is bad.
It is ... is going to ... is going from zero to something [really large] in no time.
So that quantity is infinite at this point.
That's exactly what's too bad.
So the jerk is ultimately responsible.
So that's the jerk times mass.
So the jerk is responsible for the bad behavior here.
Now here is my question.
Why is it that the jerk ...
... is really something that makes jets fly in 3-space?
Not in a plane.
It's exactly the jerk that pulls the plane out of the plane.
Why is it that jerk is something breaking mechanism?
At the same time ...
... many people ...
... well I assume most of you ...
... didn't have any experience about the jerk.
Right? So what is it that we have all that?
Something beautiful about jerk.
Something dangerous about jerk.
Something mysterious about jerk that nobody experienced before.
And we still never study jerk in calculus.
Why is it that we study in Calculus, in Calculus I velocity and acceleration?
And what do we study about them?
We usually just reconfirm the well-known, well-heard concepts.
Newton's law and ...
... the distance travelled, right?
So it is usually all about reconfirming something we already experienced ...
... we have an idea about.
And yes indeed Calculus confirms, that's right.
What a beautiful use of Calculus!
What a valuable contribution!
So the real strength of Calculus, the real power of Calculus is that ...
... from Calculus point of view there is absolutely no difference between ...
... considering velocity, acceleration or jerk.
It's just another derivative. Right?
So the question is why don't we use Calculus systematically?
And push our knowledge further.
And learn something finally about the jerk.
And understand something better about flight ...
... about mechanisms, about ...
... what breaks, what doesn't break.