Paradoxical properties of the line of pursuit in the problem of intercepting a fugitive on a horizontal plane

Background. A new approach to the construction, integration and analysis of the differential equation of the pursuit curve in the classical problem of intercepting a fugitive on the plane is considered. In the proposed wording, it is new and relevant from a practical point of view in such areas as transport, logistics, military affairs, sports events, etc. Objective. The purpose of the work is to determine the optimal angle of inclination of the straight line along which the fugitive should move in order to get as close as possible to the "life" line before being caught by the pursuer. Methods. To achieve the goal, classical methods of integrating differential equations in parametric form were used, as well as graphic and numerical tools provided by the MathCad software package. Results. The differential equation of the pursuit curve is formulated and its solution in closed parametric form is established. Its numerical analysis was carried out and the influence of the parameters (ratio of speeds of the pursuer and the fugitive) and on the behaviour of the pursuit curve was investigated. The dependence of the change in the distance along the horizontal axis at the moment of apprehension of the fugitive, depending on the magnitude of the angular coefficient of the straight line of his movement at a fixed coefficient , was analysed. Conclusions. As a result of the research, it was found that if the coefficient (that is, the speed of the pursuer significantly exceeds the speed of the fugitive), then the distance at which the latter will be detained goes to zero. On the other hand, if (that is, the speeds of the boats are equal), then the specified distance goes to infinity, that is, the fugitive will not be apprehended. The dependence of the change of the distance along the horizontal axis on the coefficient has a well-defined local maximum under the condition , which indicates that there is a certain non-zero angle of inclination of the direct flight, which enables the fugitive to achieve the maximum movement towards the border between the two countries. Thus, we observe a paradoxical phenomenon: for a successful escape, the strategy of choosing an angle of inclination of a straight line equal to zero (that is, moving along the shortest segment connecting two parallel lines) is not correct.