Teaching Calculus

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  • Nikolay Brodskiy
  • Alexander Brodsky

Part 05 (Transcript)

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So ...

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So what is it about that ...

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... orthogonal component that tells us about the shape of the curve?

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Student: [...].

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Yes, I can [...] ...

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Well I can do that if ...

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... if I think in terms of that t.

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But geometrically I don't.

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Right?

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So that's the problem.

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Geometrically you don't think about a particle moving along the curve.

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Quickly or slowly.

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Quickly here and slowly there.

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So you think about the shape.

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That's all you can see.

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So.

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So let me ...

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Let me mention ...

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Er ... remember the ...

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... stages of studying curves that I outlined some time ago geometrically.

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So we studied curves from the point of view of lines.

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Right? And that was analogous to the first degree approximation.

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But then the next stage was about circles.

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Right? Because that was the next kind of complicated ...

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... geometric object.

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And the hope is to approximate any curve by a circle.

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So having that hope in mind ...

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... can we see a circle ...

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... in algebra?

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Well no way. What we see is quadratic parabola.

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Right? So it cannot be a circle.

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It's never a circle.

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But in physics can we think of a circle?

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Can we think of a circular motion?

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Now you'll have to use some imagination.

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Imagine this curve being a road.

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And you drive along that road.

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OK?

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So you drive and ...

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... turn the wheel.

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And then of course to make this turn you have to turn the wheel somewhat.

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But then the point of imagination should be that at this moment ...

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... you  ...

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... stop turning the wheel.

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So from this moment on you fix the wheel and you don't turn it at all.

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Of course this way you will not follow the road.

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But how will you drive?

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Student: On a circle.
NB: Off the road.

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Wouldn't you follow the circle?

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So if you stop turning the wheel ...

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... you will follow that circle.

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That's how geomtry can be seen and experienced in physics.

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Student: So the orthogonal component is the radius of the circle.

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Possibly.

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Possibly.

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Well [...] we have some relevance of that orthogonal component ...

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... to the circular motion and hope that ...

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... studying that circular motion ...

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... will give us idea about what the circle is.

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And therefore give us geometry of what's going on.

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