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People are familiar with chases and escapes since their childhood when every child played the game hide-and-seek.
On January 16, 1732, the French mathematician and hydrographer Pierre Bouguer (1698--1758) read his paper before
the French Academy where he first formulated and solved the pursuit problem. Bouguer treated the case of a pirate ship
pursuing a fleeing merchant vessel that he assumed to travel along straight line with constant speed v_{m}.
The pirate ship travels at constant speed v_{p} along a curved path such that it is always moving
directly toward the merchant, that is, the velocity vector of the pirate ship points directly at the merchant vessel
at every instant of time. Bouguer's problem was to determine the equation \( y = y(x) \)
of the curved path which is called the curve or ine of pursuit.
To find the curve of pursuit for Bouguer's problem, let us denote by (x,y) the location of pirate ship
in rectangualr coordinate system at arbitrary time \( t \ge 0 . \) We assume that
initially, the pirate ship was at the origin, and the merchant vessel was at the point
\( \left( x_0 , 0 \right) . \) At time t, the merchant vessel has sailed to the
point \( \left( x_0 , v_m t \right) . \) The slope of the tangent line to the pursuit curve is
We also know that, whatever the shape of the pursuit curve, the pirate ship has sailed along it at time t by
distance \( v_p t . \) From calculus, we know taht this arc-length is also given by the integral
Since it is assumed that the pirate ship travels faster that merchant vessel (otherwise, the problem has no
solution), the constant n is less than 1. The n = 1 case is actually called tractrix
and it was consided previously. The above equation is separable
where the constant of integration is determined from the initial conditions
\( p = {\text d}y / {\text d}x =0 \) at t = 0 when x = 0. Therefore,
\( C = -n\, \ln x_0 , \) and we get
``Capture'' occurs when x = x_{0}, that is, the pirate ship pursuit curve intersects the merchant's course.
Of course, we have to assume that n is less than 1.
A. E. Dubinov, I. D. Dubinova, S. K. Saykov, The Lambert W-function and its Applications to Mathematical Problems of
Physics (in Russian), Syrov (Russia), The Russian Federal Nuclear Center, 2006, pp. 1-160.
S. G. Kazakova and E. S. Pisanova, Some Applications of the Lambert W-function to Theoretical
Physics Education.
Nahin, P.J., Chases and Escapes: The Mathematics Of Pursuit And Evasion (Princeton Puzzlers), Princeton University Press; Revised edition 2012.
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