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Lecture 2 - Motion

Lecture 2 - Motion - = ∆ = ∆ ∆ = ∆ ∆ = = ∆ 29...

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The Law of Falling Bodies and The Calculus
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The Law of Falling Bodies In a vacuum all bodies fall with the same constant acceleration
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2 1. Aristotle: natural place. Heavier=faster. 2. Albert of Saxony (12th C): 3. Nicole Oresme (14th C): 4. Nicole Oresme: 5. Leonardo da Vinci (15th C): integers 6. Galileo Galilei (16th C): v s v t s t s ∆ ∝ odd numbers s ∆ ∝
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lim 0 lim 0 t t s v t s ds v t dt v a t v dv a t dt ∆ → ∆ → = = = = = =
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( 29 ( 29 1 1. Constants 0 2. Power law 3. Sum rule 4. Product rule n n dc dt d x nx dx d dx dy x y dt dt dt d dx dy xy y x dt dt dt - = = + = + = +
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( 29 ( 29 5. Chain rule and y x x t dy dy dx dt dx dt =
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6. Special functions sin cos cos sin 1 log x x e d x x dx d x x dx d e e dx d x dx x = = - = =
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2 1 1 Quadratures: area under a circle = where 4, but = what? c r c p 2 3 0 0 0 3 2 0 2 Area under a parabola: 1 1 Area under the triangle: 2 2 Area under parabola: , 1 where , but = what? 2 y x x y x c x c = = p
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2 2 2 lim 0 Area of sliver x A x x A x x A dA x x dx ∆ →
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Unformatted text preview: = ∆ = ∆ ∆ = ∆ ∆ = = ∆ ( 29 ( 29 2 3 3 2 1 3 1 3 1 and 3 dA x dx A x C A x A x C = = +-= = ( 29 ( 29 ( 29 ( ) ( ) b a dA f x dx A f x dx A b A a f x dx = =-= ∫ ∫ ( 29 constant t dv a g dt v gdt v f t gt c v gdt gt = = = = = = + = = ∫ ∫ 2 1 2 ds v dt s vdt gt c = = = + ∫ 2 2 1 2 more generally, 1 2 a g v gt s gt v v at s s v t at = = = = + = + + 1 2 2 Score card: 1. Aristotle: wrong (except in the real world) 2. Albert of Saxony: wrong 3. Nicole Oresme: correct 4. Nicole Oresme: correct 5. Leonardo da Vinci: integers correct, but doe v s v t s t s ∝ ∝ ∝ ∆ ∝ sn't start from rest 6. Galileo Galilei: odd numbers correct, but odd s ∆ ∝...
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