ballistics (2D) Ballistics 2006-07-25 16:23:33 2009-06-12 09:59:12 Definition ballistic testing reference system 2D-projection projected motion kinematic function relative motion kinematics intercontinental ballistics ballistics applications % this is the default PlanetPhysics preamble. as your % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % making logically defined graphics % define commands here {\em Ballistics} is the study of the kinematics and dynamics of a projected motion of an object. The object in motion relative to the agency applying the force causing the beginning of the accelerated motion of the object is thus called a ``projectile''. Note however that in the presence of a gravitational field the latter also contributes to either accelerating or decelerating the projectile depending on the initial conditions, i.e. the orientation and direction of the applied force relative to gravitational forces. See also, for example, in the following movie download the \PMlinkexternal {marked effects of a Coriolis force{http://ww2010.atmos.uiuc.edu/(Gh)/guides/mtr/fw/gifs/coriolis.mov}, and more generally those of a spinning projectile on its trajectory. In many cases, a two-dimensional projection can considered to be a sufficient approximation, or a first step in attempting to define the trajectory of a projectile object in its motion relative to that of an observer, or reference system. Let us define first a horizontal axis $x$ and a vertical axis $y$ in the latter reference system of the observer. \subsection{Horizontal Motion} This will be considered first. Assuming that the drag force is insignificant, the sum of forces along the $x$ axis $\Sigma F_x$ equals zero, and so the acceleration along the axis $a_x$ is also insignificant. Thus, the velocity along the the axis $v_x$ is constant and equal to the projection velocity $V_{i,x}$. When dealing with a constant velocity motion the following kinematic function is relevant: \[ x=x_i+v_x \cdot \Delta t \rightarrow x=x_i+ \Delta t\cdot V_i \cos \alpha \] ($x_i$-the initial position, $V_x$-the velocity along the axis, $\Delta t$-the duration) \subsection{Vertical Motion:} The motion along the $y$ axis is under the effect of gravity $(mg)$ when $m$ is the object's mass and $g$ is the the free fall acceleration on earth. $\rightarrow \Sigma F_y=mg$ Because the force along the $y$ axis is constant, the motion along the axis is evenly accelerated, and so the position of the object as a function of time $y(t)$ is: \[ y= y_i+\Delta t \cdot V_{i,y} + 0.5g\Delta t^2 \rightarrow y= y_i+\Delta t \cdot V_i\sin\alpha + 0.5g\Delta t^2 \] \subsection{2D Motion:} The merging of the functions of place-time $x(t),y(t)$ produces the route equation $y(x)$: \[ y= y_i + (x-x_i) V_i \tan\alpha + \frac {g(x-x_i)^2}{2v_i^2 \cos^2\alpha} \] \textbf{Remarks:} Ballistics has also been a subject of great interest to both army and navy engineers that wished to improve the performance of various types of guns and succeded in doing so by applying physical principles and mathematics. One such early apllications of ballistics is discussed in the Old Testament, in the widely known story about David and Golliath. Other earlier examples of the use of ballistics in battles and sieges were the the ancient use of catapults in Roman conquests. Perhaps even older was the use of gun powder propelled rockets by the ancient Chinese entertainers, and also in hunting by hurtling stones spears and arrows by primitive {\em H. sapiens} bands or tribes, and earlier still by hominins and hominids; the latter, the Chinese and the Romans however did not obviously benefit from the physical science of ballistics that began only with Galileo Galilei's foundation of experimental, classical kinematics, fully dveloped later by Newton's foundation of dynamics and Netwonian calculus. Ballistics is also of considerable importance today in forensic science that also uses such physical principles combined with actual experimental ballistic testing. Ballistic considerations were also of the essence during the Cuban missile crisis, the earlier period of the Cold War, and in all earlier, `hot wars'. Intercontinental ballistics, satellite launches and NASA's interplanetary programs also make intensive use of ballistics, but in its more sophisticated forms; in such cases, 2D projections for ballistic calculations are obviously insufficient.