# UMass Amherst KIN 430 - differentiation-integration (9 pages)

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## differentiation-integration

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- Pages:
- 9
- School:
- University of Massachusetts Amherst
- Course:
- Kin 430 - Biomechanics

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Differentiation Integration Differentiation Mathematical process for finding the instantaneous rate of change of a variable related to finding the slope of a line position velocity acceleration by means of differentiation Analytical Numerical Graphical Integration Mathematical process for finding the total area under a curve basically a summation process The rate of change of one variable pos vel with respect to another time velocity is calculated from position and time acceleration is calculated from vel and time Analytical Numerical Graphical Copyright 2014 University of Massachusetts Amherst Derivative is the slope of a line s2 rise Average vs Instantaneous pos s1 or vel Basic kinematic formulas give average values only run e g tf ti time There will be situations were you need to know the instantaneous velocity and acceleration Vel slope of the pos vs time curve Acc slope of the vel vs time curve 1 Analytically instantaneous Numerically close to instantaneous Determine the slope of the tangent line Determine the slope of the secant line Position Position tangent t secant ti 1 Time finite difference method Time s Pos m Vel m s 0 0 3 20 A numerical technique that provides an approximation to the true instantaneous derivative ti ti 1 Time First finite difference method 3pt Time s Pos m Vel m s 0 0 3 20 0 1 5 80 52 0 0 2 13 60 48 0 0 3 15 40 19 0 0 1 5 80 0 2 13 60 0 3 15 40 0 4 9 80 0 4 9 80 39 0 0 5 7 60 0 5 7 60 Example Use for points 2 through n 1 2 First forward difference method 2pt First backward difference method 2pt Time s Pos m Vel m s Time s Pos m Vel m s 0 0 3 20 26 0 0 0 3 20 26 0 0 1 5 80 52 0 0 1 5 80 52 0 0 2 13 60 48 0 0 2 13 60 48 0 0 3 15 40 19 0 0 3 15 40 19 0 0 4 9 80 39 0 0 4 9 80 39 0 0 5 7 60 0 5 7 60 22 0 Use for point 1 Use for point n Calculating acceleration Graphical Differentiation Time s Pos m Vel m s Acc m s2 0 0 3 20 26 0 260 0 1 5 80 52 0 110 0 2 13 60 48 0 355 0 3 15 40 19 0 435 0 4 9 80 39 0 15 0 5 7 60 22 0 170 Sometimes it may be

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