Pursuit CurvesMolly SeverdiaMay 15, 2008Molly Severdia Pursuit CurvesAssumptionsIAt t = 0, merchant at(x0, 0), pirate at (0, 0).IMerchant’s speed is Vm.IPirate’s speed is Vp.IMerchant travels alongvertical line x = x0.IAt time t ≥ 0, pirate at(x, y).xyx0− xVmt − y(x, y)x0(x0, Vmt)y = y(x)Figure: Geometry of pirate pursuitMolly Severdia Pursuit Curvesdydx=Vmt − yx0− xVpt =Zx0s1 +dydz2dzxyx0− xVmt − y(x, y)x0(x0, Vmt)y = y(x)Figure: Geometry of pirate pursuitMolly Severdia Pursuit CurvesDifferential Equation for Pirate Pursuit(x − x0)dpdx= −nq1 + p2(x)n =VmVp, p(x) =dydxMolly Severdia Pursuit CurvesSeparable Equationdpp1 + p2=−n dxx − x0ln(p +p1 + p2) + C = −n ln(x0− x)dydx=12"1 −xx0−n−1 −xx0n#y(x) =12(x − x0)(1 − x/x0)n1 + n−(1 − x/x0)−n1 − n+n1 − n2x0Molly Severdia Pursuit CurvesSeparable Equationdpp1 + p2=−n dxx − x0ln(p +p1 + p2) + C = −n ln(x0− x)dydx=12"1 −xx0−n−1 −xx0n#y(x) =12(x − x0)(1 − x/x0)n1 + n−(1 − x/x0)−n1 − n+n1 − n2x0Molly Severdia Pursuit CurvesResults0 5 10 1500.511.522.533.54x-axisy(x)n=0.3Figure: Results using ode45Molly Severdia Pursuit CurvesCircular Pursuit”A dog at the center of a circular pond makes straight for a duckwhich is swimming [counterclockwise] along the edge of the pond.If the rate of swimming of the dog is to the rate of swimming ofthe duck as n : 1, determine the equation of the curve of pursuit...”Molly Severdia Pursuit CurvesGeneric CasexyduckhoundOh(t)d(t)ρρρ(t)d(t) = h( t) + ρρρ(t)d(t) = xd(t) + iyd(t) h(t) = xh(t) + iyh(t)Molly Severdia Pursuit CurvesDuckIDuck’s position vector given byd(t) = xd(t) + iyd(t)IDuck’s velocity vector given bydd(t)dt=dxddt+ idyddtIDuck’s speed isdd(t)dt=sdxddt2+dyddt2Molly Severdia Pursuit CurvesDuckIDuck’s position vector given byd(t) = xd(t) + iyd(t)IDuck’s velocity vector given bydd(t)dt=dxddt+ idyddtIDuck’s speed isdd(t)dt=sdxddt2+dyddt2Molly Severdia Pursuit CurvesDuckIDuck’s position vector given byd(t) = xd(t) + iyd(t)IDuck’s velocity vector given bydd(t)dt=dxddt+ idyddtIDuck’s speed isdd(t)dt=sdxddt2+dyddt2Molly Severdia Pursuit CurvesHoundIHound’s position vector given byh(t) = xh(t) + iyh(t)IHound’s velocity vector is given bydh(t)dt=dh(t)dt·ρρρ(t)|ρρρ(t)|(1)IHound’s speed is n times that of the duck,dh(t)dt= nsdxddt2+dyddt2Molly Severdia Pursuit CurvesHoundIHound’s position vector given byh(t) = xh(t) + iyh(t)IHound’s velocity vector is given bydh(t)dt=dh(t)dt·ρρρ(t)|ρρρ(t)|(1)IHound’s speed is n times that of the duck,dh(t)dt= nsdxddt2+dyddt2Molly Severdia Pursuit CurvesHoundIHound’s position vector given byh(t) = xh(t) + iyh(t)IHound’s velocity vector is given bydh(t)dt=dh(t)dt·ρρρ(t)|ρρρ(t)|(1)IHound’s speed is n times that of the duck,dh(t)dt= nsdxddt2+dyddt2Molly Severdia Pursuit CurvesIEquation (1) becomesdh(t)dt= nsdxddt2+dyddt2·d(t) − h(t)|d(t) − h(t)|IIn Cartesian Coordinates,dxhdt+idyhdt= nsdxddt2+dyddt2·(xd− xh) + i(yd− yh)p(xd− xh)2+ (yd− yh)2IEquating real and imaginary parts leads to...Molly Severdia Pursuit CurvesIEquation (1) becomesdh(t)dt= nsdxddt2+dyddt2·d(t) − h(t)|d(t) − h(t)|IIn Cartesian Coordinates,dxhdt+idyhdt= nsdxddt2+dyddt2·(xd− xh) + i(yd− yh)p(xd− xh)2+ (yd− yh)2IEquating real and imaginary parts leads to...Molly Severdia Pursuit CurvesIEquation (1) becomesdh(t)dt= nsdxddt2+dyddt2·d(t) − h(t)|d(t) − h(t)|IIn Cartesian Coordinates,dxhdt+idyhdt= nsdxddt2+dyddt2·(xd− xh) + i(yd− yh)p(xd− xh)2+ (yd− yh)2IEquating real and imaginary parts leads to...Molly Severdia Pursuit CurvesEquations for General Pursuitdxhdt= nsdxddt2+dyddt2·xd− xhp(xd− xh)2+ (yd− yh)2dyhdt= nsdxddt2+dyddt2·yd− yhp(xd− xh)2+ (yd− yh)2Molly Severdia Pursuit CurvesIIf the duck swims counterclockwise around a unit circle,xd(t) = c os(t) , yd(t) = sin(t).IAlso,nsdxddt2+dyddt2= nqsin2(t) + cos2(t) = nMolly Severdia Pursuit CurvesIIf the duck swims counterclockwise around a unit circle,xd(t) = cos(t) , yd(t) = sin(t).IAlso,nsdxddt2+dyddt2= nqsin2(t) + cos2(t) = nMolly Severdia Pursuit CurvesCircle Pursuitdxhdt= ncos(t) − xhp(cos(t) − xh)2+ (sin(t) − yh)2dyhdt= nsin(t) − yhp(cos(t) − xh)2+ (sin(t) − yh)2Molly Severdia Pursuit Curves−1 −0.5 0 0.5 1−1−0.500.51n = 0.3−1 −0.5 0 0.5 1−1−0.500.51n = 0.3Molly Severdia Pursuit Curves−1 −0.5 0 0.5 1−1−0.500.51n = 0.5−1 −0.5 0 0.5 1−1−0.500.51n = 0.5Molly Severdia Pursuit Curves−1 −0.5 0 0.5 1−1−0.500.51n = 0.2−1 −0.5 0 0.5 1−1−0.500.51n = 0.7Molly Severdia Pursuit Curvesxy(x, y)x0duck(a, 0)aρωθMolly Severdia Pursuit CurvesIEquation of tangent line:y cos(ω) − x sin(ω) = −a sin(ω − θ)IEquation of normal line:x cos(ω) + y sin(ω) = a cos(ω − θ) − ρMolly Severdia Pursuit CurvesDifferentiate tangent linedxdθsin(ω)−dydθcos(ω)+dωdθ[x cos(ω)+y sin(ω)] = a cos(ω−θ)dωdθ− 1ρdωdθ= a cos(ω − θ)Molly Severdia Pursuit CurvesDifferentiate tangent linedxdθsin(ω)−dydθcos(ω)+dωdθ[x cos(ω)+y sin(ω)] = a cos(ω−θ)dωdθ− 1ρdωdθ= a cos(ω − θ)Molly Severdia Pursuit CurvesDifferentiate normal linedxdθcos(ω) − x sin(ω)dωdθ+dydθsin(ω) + y cos(ω)dωdθ= −a sin(ω − θ)dωdθ− 1−dρdθdρdθ= a[sin(ω − θ) − n]Molly Severdia Pursuit CurvesDifferentiate normal linedxdθcos(ω) − x sin(ω)dωdθ+dydθsin(ω) + y cos(ω)dωdθ= −a sin(ω − θ)dωdθ− 1−dρdθdρdθ= a[sin(ω − θ) − n]Molly Severdia Pursuit Curvesρdωdθ= a cos(ω − θ)dρdθ= a[sin(ω − θ) − n]φ = ω − θdωdθ=dφdθ+ 1ρd2ρdθ2+ aρ cos(φ) = a2cos2(φ)dρdθ= a sin(φ) − anMolly Severdia Pursuit Curvesρd2ρdθ2+ aρ cos(φ) = a2cos2(φ)dρdθ= a sin(φ) − anxyρaRduck’s position hound’s limit cycleIlimθ→∞ρ = cIdρdθ=d2ρdθ2= 0IAs θ → ∞, ρ = a cos(φ)IAs θ → ∞, sin(φ) = nMolly Severdia Pursuit CurvesAs θ → ∞...aρρa= a2[1 − sin2(φ)] = a2(1 −
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