Rectilinear Motion. Velocity

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Elfreda Fitzgerald
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1 Rectilinear Motion Motion along a line Needed three things. Zero. Positive 3. Units This was our coordinate system Motion was a vector since it had magnitude and direction Average velocity Instantaneous velocity Velocity v avg v s = = t s t s lim = t 0 t = s t ds dt = s& Work backwards s = vdt Slope of position graph = velocity Area under velocity graph = change in position

2 Average acceleration Instantaneous acceleration Acceleration a a avg v v = = t t v lim = t 0 t = dv dt v t = v& = && s Work backwards v = adt Slope of velocity graph = acceleration Area under acceleration graph = change in velocity 3 Example From Lec 3- Differentiate Position (ft) Time (sec) s = 3 6t t v = t 3t a = 6t What if we have discrete data instead? Integrate Velocity (ft/sec) Acceleration (ft/sec/sec) Time (sec) Time (sec) 4

3 New Example During one of the EF 0 lectures, we set up a radar gun and recorded the horizontal (left and right) velocity of Dr. Raman. The first measurements are shown at right. Screen Podium 0 0 ft 0 ft Radar Gun Dr. Raman's Velocity Time Velocity Sec ft/sec We want to create the s-t, v-t, and a-t diagrams using Matlab. 5 Enter time - velocity data. "Hardwire" the data. Set up s-t, v-t, and a-t plots in vertical order. Use subplots. Put in title and axis labels. Plot the time - velocity data. Set axis limits. Determine the time - position data. Figure out to numerically integrate (trapezoids sound good). Determine the time - acceleration data. Figure out to numerically differentiate. Curve fit all three plots. Saw this in PS -7. Our Plan 6

4 trapz Trapezoidal numerical integration Z = trapz(y) Z = trapz(x,y) Z = trapz(y) computes an approximation of the integral of Y via the trapezoidal method (with unit spacing). To compute the integral for spacing other than one, multiply Z by the spacing increment. If Y is a vector, trapz(y) is the integral of Y. If Y is a matrix,trapz(y) is a row vector with the integral over each column. Z = trapz(x,y) computes the integral of Y with respect to X using trapezoidal integration. 7 cumtrapz Cumulative trapezoidal numerical integration Z = cumtrapz(y) Z = cumtrapz(x,y) Z = cumtrapz(y) computes an approximation of the cumulative integral of Y via the trapezoidal method with unit spacing. To compute the integral with other than unit spacing, multiply Z by the spacing increment. For vectors, cumtrapz(y) is a vector containing the cumulative integral of Y. For matrices, cumtrapz(y) is a matrix the same size as Y with the cumulative integral over each column. Z = cumtrapz(x,y) computes the cumulative integral of Y with respect to X 8 using trapezoidal integration. X and Y must be vectors of the same length,

5 diff Differences and approximate derivatives Y = diff(x) Y = diff(x,n) Y = diff(x) calculates differences between adjacent elements of X. If X is a vector, then diff(x) returns a vector, one element shorter than X, of differences between adjacent elements: [X()-X() X(3)-X()... X(n)-X(n-)] If X is a matrix, then diff(x) returns a matrix of row differences: [X(:m,:)-X(:m-,:)] Y = diff(x,n) applies diff recursively n times, resulting in the nth difference. Thus, diff(x,) is the same as diff(diff(x)). The quantity diff(y)./ diff(x) is an approximate derivative. 9 polyder Polynomial derivative k = polyder(p) k = polyder(a,b) [q,d] = polyder(b,a) The polyder function calculates the derivative of polynomials, polynomial products, and polynomial quotients. The operands a, b, and p are vectors whose elements are the coefficients of a polynomial in descending powers. k = polyder(p) returns the derivative of the polynomial p. k = polyder(a,b) returns the derivative of the product of the polynomials a and b. [q,d] = polyder(b,a) returns the numerator q and denominator d of the derivative of the polynomial quotient b/a. 0

6 polyint Integrate polynomial analytically polyint(p,k) polyint(p) polyint(p,k) returns a polynomial representing the integral of polynomial p, using a scalar constant of integration k. polyint(p) assumes a constant of integration k=0. quad, quad8 Numerically evaluate integral, adaptive Simpson quadrature Note The quad8 function, which implemented a higher order method, is obsolete. The quadl function is its recommended replacement. q = quad(fun,a,b) q = quad(fun,a,b,tol) Quadrature is a numerical method used to find the area under the graph of a function, that is, to compute a definite integral. q = quad(fun,a,b) approximates the integral of function fun from a to b to within an error of 0-6 using recursive adaptive Simpson quadrature. fun accepts a vector x and returns a vector y, the function fun evaluated at each element of x. q = quad(fun,a,b,tol) uses an absolute error tolerance tol instead of the default which is.0e-6. The function quadl may be more efficient with high accuracies and smooth integrands.

7 Summary - Discrete Data Integration - find area under curve. trapz and cumtrapz - trapezoid rule. Integration "smoothes" errors in the data. Use to go from acceleration to velocity. Use to go from velocity to position. How do we handle s 0 = 0 and/or v 0 = 0? Differentiation. Use the diff command. : diff(y)./ diff(x) differentiation "amplifies" errors in the data. Use to go from position to velocity. Use to go from velocity to acceleration. We "lose" information. Why? 3 Summary - Polynomials Use to curve fit discrete data. Use polyfit and polyval. Curve fitting also "smoothes" errors in the data. Integration - find area under curve. polyint - integrates the coefficients. Use to go from acceleration to velocity. Use to go from velocity to position. Again, how do we handle s 0 = 0 and/or v 0 = 0? Differentiation. polyder - differentiates the coefficients. Use to go from position to velocity. Use to go from velocity to acceleration. 4

Week 10 TENTATIVE SCHEDULE NUMERICAL APPROXIMATION LECTURE # 10

Week 10 TENTATIVE SCHEDULE NUMERICAL APPROXIMATION LECTURE # 10 TENTATIVE SCHEDULE INTRODUCTION TO SCIENTIFIC & ENG. COMPUTING BIL 108E, CRN 44448 Week.9_ Polynomials Examples Week.10_ Application of curve fitting : Approximation & Inregration Dr. Feyzi HAZNEDAROĞLU