Analytical Calculation Of Trajectories Using A Power Law For The Drag Coefficient Variation With Mach Number
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For hand calculations and for the fast computation of direct fire trajectories two analytical solutions of the point mass equation of motion in stationary air are presented, which are based on the power law for the drag coefficient variation with Mach number cD = CMa-m. The first quadrature solution provides time of flight τ, horizontal downrange distance x and height y as functions of the angle of flight ϕ. The second closed from solution yields ϕ (x), τ (x), y (x) and is applicable to weakly curved trajectories. Using a moving co-ordinate system, both analytical solutions can also be applied to situations with wind. Keywords: point mass trajectory, analytical solution, direct fire, drag coefficient, Mach number, wind. 1 Introduction For the precise computation of trajectories numerical methods are required and available, which take the variation of drag coefficient with Mach number into account as well as the change of air pressure and temperature along the trajectory. Wind can also be considered in the equation of motion and other side effects may be accounted for. There seams to exist no longer any need for analytical solutions. However, for direct fire applications with short distances and moving targets, highly sophisticated computer programmes are prohibitively time consuming. Not extremely but sufficiently accurate analytical solutions are more appropriate in such cases. Also for hand calculations for small arms using a pocket computer, analytical solutions are very useful. In wide regions of the
point mass trajectory, analytical solution, direct fire, drag coefficient, Mach number, wind.