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19 - Problem The velocity of a falling skydiver is given by...

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3/24/2010 1 Problem: The velocity of a falling skydiver is given by the equation below. Assuming that m = 68.1 kg and c d = 0.25 kg/m, how far will the skydiver fall between t = 4 seconds and t = 7 seconds? t m gc c gm t v d d tanh ) ( Ideal solution: 4 cosh ln 7 cosh ln cosh ln tanh ) ( 7 7 4 7 4 m gc c m m gc c m t m gc c m dt t m gc c gm dt t v s d d d d d d d d Analytic solutions are best when they are possible: Having a closed form solution can be useful Solution is not subject to inaccuracies inherent in numerical methods m 5867 . 119 4 Matlab Solution: Code Required: cd = 0.25; m = 68.1; g = 9.81; v = @(t) sqrt(g * m / cd) * tanh(sqrt(g * cd/m)*t); s = quad (v, 4, 7); % integrate v(t) for t from 4 to 7 fprintf (‘The skydiver will fall %f m\n’, s); In this case the result of 119.5867 m is exact to four decimal places Function quad performs numerical integration Numerical integration only approximates the true value of an integral (but the approximation can be very good and exact in some cases).

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3/24/2010 2 Function quad: Usage: answer = quad (func, a, b, tol, trace, p1, p2 …); func = function to be integrated should accept vectors (i.e. dot operators should be used as required) may have parameters beyond x (see p1 p2 below) (see p1, p2, … below) a , b = limits of integration tol = desired absolute error tolerance (default = 10 6 ) trace = display option (nonzero values cause information to be displayed) p1 , p2 , …. = parameters for function
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19 - Problem The velocity of a falling skydiver is given by...

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