# Pursuit curves and arc length question

I am studying pursuit curves where a fast pirate ship which pursues a heavily laden treasure ship which tracks along a straight line. The ratio of the speeds of the ships is r > 1 (which is fixed) and the pirate captain spies the treasure ship initially at a distance d km away.

There s a step within the calculations thatI am failing to understand:

I do not understand why or how the speed is equal to r(dY/dt)

Any help is greatly appreciated! thank you

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What is $Y$?${}{}{}$ – Américo Tavares Apr 12 '13 at 12:07

## 1 Answer

Just read the same article Solving the basic pursuit problem by Peter Hagstrom. The trade ship goes straight along the $Y$-axis (see the pic.) so its speed is $dy/dt$. The pirate moves in a plane, both its coordinates change, that's why $ds/dt$. And it is $r$ times more than the trade ship speed. As for me I study the asymptotics of the length of the pursuit curve when $r\gg1$.

Yuri V. Osipov prof., Chair of Applied Math., Moscow University of Civil Engineering> Russia

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