Lecture 2 - Exact (Analytic) Solution Newtons Second Law y,...

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Newton’s Second Law Exact (Analytic) Solution 2 d 2 d v m c g dt dv v c mg dt dv m - = - = Exact Solution = t m gc c mg t v d d tanh ) ( y, v, a From your Diff Eqn course: IC: v(t=0)=0 mg v 2
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2 2 2 [ ] 2 1 1 1 d d d d d d d d c m dv dv dv dt dt g dt v dv dt gm c g gm v v c gm c c g m m m v c v c m c = = = - + = - + - - Method of separation of variables for the ODE ( ) ln( ) [-ln ] 2 d d d d gm gm gm c c c v v t c c m - + + = + IC: v(t=0)=0 => c =0… then simplify using definiton of tanh(x)
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Re= V d / ν 2 ( ) d c v dv g v dt m = - Diff Eqn kinematic viscosity
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Numerical Method 0 1 1 ( ) ( ) lim t i i i i dv v dt t v v t v t t t t ∆ → + + = - = - If c d = c d (v) const Solve the ODE numerically! Q: how to COMPUTE dv/dt numerically? Good if t is small v t From Calculus: 2 d dv c g v dt m = - Diff Eqn
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Finite difference (Euler’s) method Numerical (Approximate) Solution 2 i d i 1 i i 1 i i 1 i i 1 i t v m c g t t t v t v t t t v t v t v dt dv ) ( ) ( ) ( ) ( ) ( - = - - - - = 2245 + + + + Numerical Solution ) ( ) ( ) ( ) ( i 1 i 2 i d i 1 i t t t v
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Lecture 2 - Exact (Analytic) Solution Newtons Second Law y,...

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