Расчет длины траектории для задачи преследования
A classic pursuit problem is studied in which two material points - a Pursuer and a Pursued - move in a plane at constant speeds. The velocity vector of the Pursued does not change its direction and the velocity vector of the Pursuer turns and always aims at the Pursued. If the Pursuer moves at a higher speed, it will overtake the Pursued for any initial angle between velocity vectors.
The shape of the mechanical trajectory is established. The trajectory line rotates about the origin so that at the final meeting point the tangent line to the motion trajectory always coincides with the velocity vector of the Pursued. The two-parameter integral for the length of the pursuit curve is considered, its asymptotics up to the forth term is calculated under the assumption that the speed of the Pursuer is much greater than the speed of the Pursued. Rapid convergence of the asymptotics to the integral for the trajectory length is provided by the absence of the first and the third terms of the asymptotic expansion. Numerical computation of the trajectory length is compared with the asymptotic formulas. Calculations show that the resulting asymptotics is a good approximation of the integral for the trajectory length, and the fourth term in the asymptotic formulas significantly improves the approximation.