07-Nonlinear-Equations-4UP - Applications Zero-nding Iterative methods Applications Zero-nding Iterative methods Nonlinear Equations Nonlinear Equations

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

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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 θ V 0 y x Parameters: g = acceleration of gravity ( ms - 2 ) : known V 0 = initial speed ( ms - 1 ) : known d = distance to target ( m ) : known θ = required elevation (radians): unknown Determine elevation θ needed to hit target using known values of parameters V 0 , 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 Newton’s 2nd law x ( t ) = 0, x ( 0 ) = 0, x ( 0 ) = V 0 cos θ y ( t ) = - g , y ( 0 ) = 0, y ( 0 ) = V 0 sin θ ODE system integrates directly to yield x ( t ) = ( V 0 cos θ ) t y ( t ) = ( V 0 sin θ ) 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 ) = 0 If y ( T ) = 0, then T = 0 or T = 2 V 0 sin θ g Reject T = 0, so we must have x ( T ) = ( V 0 cos θ ) 2 V 0 sin θ g = d Zero-finding problem: find elevation θ * such that f ( θ * ) = 0, where f ( θ ) : = 2 sin θ cos θ - dg V 2 0 c D. Aruliah (UOIT) Nonlinear Equations MATH 2070U 6 / 23 Applications Zero-finding Iterative methods Remarks: Range of a cannonball d θ V 0 y x Idealisation: no air resistance Solution does not exist if dg V 2 0 > 1