Preface

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This is a tutorial made solely for the purpose of education and it was designed for students taking Applied Math 0330. It is primarily for students who have very little experience or have never used Mathematica before and would like to learn more of the basics for this computer algebra system. As a friendly reminder, don't forget to clear variables in use and/or the kernel.

Finally, the commands in this tutorial are all written in bold black font, while Mathematica output is in normal font. This means that you can copy and paste all commands into Mathematica, change the parameters and run them. You, as the user, are free to use the scripts for your needs to learn the Mathematica program, and have the right to distribute this tutorial and refer to this tutorial as long as this tutorial is accredited appropriately.

Pursuit

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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 where s is a dummy variable of integration. Using the expression for the derivative, we find which, if we let \( {\text d}y /{\text d} x = p(x) , \) becomes Differentiating the above equation with respect to x, we arrive at Using some algebra, we simplify the above equation to the following: 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 Integrating both sides, we obtain 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 Upon some pencil pushing, we arrive at which yields If we temporarily set \( q = p + \sqrt{1+p^2} \quad \Longrightarrow \quad p = \left( q^2 -1 \right) /(2q) , \) then Once more integration yields Factoring out the common term, we have Now we determine the value of constant of integration from the initial condition \( y=0 \) when x = 0: Substituting into the formula for y(x), we get ``Capture'' occurs when x = x₀, 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), S§rov (Russia), The Russian Federal Nuclear Center, 2006, pp. 1-160.
Some Applications of the Lambert W-function to Theoretical Physics Education S. G. Kazakova and E. S. Pisanova