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Transcript

Curve of Pursuit

The Work

started with the slope of the tangent line to the curve of pursuit which was when vessel A was moving directly toward vessel B dy/dx=(y-Beta(t))/(x-b). With given speeds we knew the distance traveled at t was alpha(t). Gave us the equation alpha*t=∫_0^x√ (1+[y'(u)]^2 )du (arclength formula). solve for t for both of these equations.

The Basics

My main Aim

The general and simplest problem of a curve of pursuit is finding the curve along which a vessel moves in pursuing another vessel that flees along a straight line, while assuming that the speeds of both vessels are constant.

My goal was to find the Locus of points P, which is to find y as a function of x, also to find the location where vessel A intercepts vessel B when alpha>Beta as well as showing if vessel A will ever reach vessel B when Alpha=Beta.

Origin/Background

work cont..

When trying to determine the path of an pursuer chasing its prey or target, the path is considered the curve of pursuit. In the 1730’s these problems were being analyzed using methods of calculus but two centuries before Leonardo da Vinci had began scratching the surface with the problems but left the thought incomplete.

set both equations that you found for t equal to each other. 1/alpha ∫_0^x√(1+[y'(u)]^2 ) du = -y/Beta -(b+x)/Beta dy/dx. After that, differentiate both sides of the equation with respect to x and derived the first order equation which resulted in (x-b)dw/dx=-Beta/ Alpha √(1+w^2 )

More work..

w set to equal dy/dx. Use separation of variable along with initial conditions x=0 and w=dy/dx=0 when t=0, the equation you get is ( dy)/(dx)=w=1/2[(1-x/b )^(-Beta)/alpha)- (1- x/( b) )^(Beta/alpha) ].

Reaching my goals

With finding the location where vessel A intercepts vessel B if alpha > Beta is b/2(b(alpha*Beta)/alpha^2-Beta^2. Finally, when alpha = Beta does vessel A ever reach vessel B, no it doesn't since x=b and the curve of pursuit equation goes to infinity with any arbitrary constant of b also by the limit x that approaches any decreasing sequence to positive numbers converging to zero for (A -B) = b/2. Vessel A will always come up short.

cont...

vessel A travels faster than vessel B, alpha> Beta the initial conditions x=0, y=0 at t=0 used those to derive the curve of pursuit which was y = b/2[1-x/b)^(Beta/Alpha+1)/(Beta/alpha+1) - (1-x/b)^(-Beta/alpha+1)/(-Beta/alpha+1)]+ b*alpha*Beta/(alpha^2-Beta^2 ).

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