Physical model, theoretical aspects and applications of the flight of a ball in the atmosphere. Part II: Theoretical aspects in the case of vertical angular frequency and applications

Abstract

Communicated by H. NeunzertIf a ball is viewed as a rigid body, its flight in the atmosphere can be described by a system of six ordinary differential equations, which has been derived in the first part of this paper.In this following second part, the theoretical aspects such as the curvature of the orbit and certain velocity functions will be investigated in the case of the vertical angular frequency of the rotating ball, in which the differential equations reduce to a planar dynamical system. This system turns out to be not explicitly solvable.The solutions of the corresponding ordinary or boundary value problems, computed numerically, are used to treat certain problems in international ball games, for example, the maximum and minimum velocities of a volleyball service.

Effect of altitude on football performance

“…Although most of the focus in the medical literature is on the physiological effect of altitude on performance, the physical effects of altitude (Fuchs 1991, 1995) may actually be equally or even more important, especially for sports for which success depends substantially on a high degree of technical skill with a ball, as opposed to aerobic power and endurance. In the United States, it has been often discussed that altitude resident teams in Denver, Colorado at 1609 m have much better home records than they do away records.…”

Physical model, theoretical aspects and applications of the flight of a ball in the atmosphere. Part III: Theory in the case of vertical angular frequency and in the general spin‐case

“…The results will exceed the ones in Part I1 and they will be valid for the 'general oIE-case', not only for 'no shooting star-balls' as in [6]. The formulae in the shooting star-case turn out to be much more complicated since it is not possible to shift the independent variable such that it equals zero at the maximum point of the trajectory.…”

“…The qualitative results about the velocity functions u, u, and uE which have been achieved in [6, section 2.33 with a lot of theoretical effort, can be worked out much easier using a special inner transformation according to definition 2.3. The results will exceed the ones in Part I1 and they will be valid for the 'general oIE-case', not only for 'no shooting star-balls' as in [6]. The formulae in the shooting star-case turn out to be much more complicated since it is not possible to shift the independent variable such that it equals zero at the maximum point of the trajectory.…”

“…Since the splitting of the horizontal component of the velocity into 0 , and uy is clear ( [6], formula (2.13)) and since vE is given by ug = u' -u:, the system can now be viewed by investigating (u, v,) only (see also (DI)) in the 'running' vertical spin-case.…”

“…( is part of what is going to be treated here. Part I1 of this paper already deals with the case of vertical angular frequency in section 2 and we will add up some (three) facts to theoretical aspects here: First, the equivalence of the system to a very simple scalar equation, second, the explicit solution of the system by a special inner transformation, which contains the shooting star-case contrary to [6], and third, a dynamical constant, which characterizes the three possible types of orbits. Let us assume for the following that…”