Paper 3, Section I, C

Vector Calculus | Part IA, 2010

A curve in two dimensions is defined by the parameterised Cartesian coordinates

x(u)=aebucosu,y(u)=aebusinux(u)=a e^{b u} \cos u, \quad y(u)=a e^{b u} \sin u

where the constants a,b>0a, b>0. Sketch the curve segment corresponding to the range 0u3π0 \leqslant u \leqslant 3 \pi. What is the length of the curve segment between the points (x(0),y(0))(x(0), y(0)) and (x(U),y(U))(x(U), y(U)), as a function of UU ?

A geometrically sensitive ant walks along the curve with varying speed κ(u)1\kappa(u)^{-1}, where κ(u)\kappa(u) is the curvature at the point corresponding to parameter uu. Find the time taken by the ant to walk from (x(2nπ),y(2nπ))(x(2 n \pi), y(2 n \pi)) to (x(2(n+1)π),y(2(n+1)π))(x(2(n+1) \pi), y(2(n+1) \pi)), where nn is a positive integer, and hence verify that this time is independent of nn.

[You may quote without proof the formula κ(u)=x(u)y(u)y(u)x(u)((x(u))2+(y(u))2)3/2.\kappa(u)=\frac{\left|x^{\prime}(u) y^{\prime \prime}(u)-y^{\prime}(u) x^{\prime \prime}(u)\right|}{\left(\left(x^{\prime}(u)\right)^{2}+\left(y^{\prime}(u)\right)^{2}\right)^{3 / 2}} . ]

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