07-Nonlinear-Equations-4UP

07-Nonlinear-Equations-4UP - Applications Zero-finding...

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Unformatted text preview: Applications Zero-finding Iterative methods Nonlinear Equations Dhavide Aruliah UOIT MATH 2070U c D. Aruliah (UOIT) Nonlinear Equations MATH 2070U 1 / 23 Applications Zero-finding Iterative methods Nonlinear Equations 1 Applications of nonlinear equations 2 Finding zeros of nonlinear equations 3 Iterative methods for solving nonlinear equations c D. Aruliah (UOIT) Nonlinear Equations MATH 2070U 2 / 23 Applications Zero-finding Iterative methods Example: Range of a cannonball To what elevation should the cannon be raised to hit the target? d V y x Parameters: g = acceleration of gravity ( ms- 2 ) : known V = initial speed ( ms- 1 ) : known d = distance to target ( m ) : known = required elevation (radians): unknown Determine elevation needed to hit target using known values of parameters V , d , and g c D. Aruliah (UOIT) Nonlinear Equations MATH 2070U 4 / 23 Applications Zero-finding Iterative methods Example: Range of a cannonball Coordinates of cannonball at time t are ( x ( t ) , y ( t )) Motion of cannonball determined by Newtons 2nd law x 00 ( t ) = 0, x ( ) = 0, x ( ) = V cos y 00 ( t ) =- g , y ( ) = 0, y ( ) = V sin ODE system integrates directly to yield x ( t ) = ( V cos ) t y ( t ) = ( V sin ) t- 1 2 gt 2 c D. Aruliah (UOIT) Nonlinear Equations MATH 2070U 5 / 23 Applications Zero-finding Iterative methods Example: Range of a cannonball Want to find time T such that x ( T ) = d and y ( T ) = If y ( T ) = 0, then T = 0 or T = 2 V sin g Reject T = 0, so we must have x ( T ) = ( V cos ) 2 V sin g = d Zero-finding problem: find elevation * such that f ( * ) = 0, where f ( ) : = 2 sin cos - dg V 2 c D. Aruliah (UOIT) Nonlinear Equations MATH 2070U 6 / 23 Applications Zero-finding Iterative methods Remarks: Range of a cannonball...
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This note was uploaded on 02/23/2010 for the course MATH 2070 taught by Professor Aruliahdhavidhe during the Spring '10 term at UOIT.

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07-Nonlinear-Equations-4UP - Applications Zero-finding...

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