ballistics (2D)Ballistics2006-07-25 16:23:332009-06-12 09:25:29Definitionballistic testingballistic applications% this is the default PlanetPhysics preamble. as your knowledge
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% define commands here{\em Ballistics} is the study of the dynamics and kinematics of a projected object.
Let's define first a horizontal axis $x$ and a vertical axis $y$.
\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 $y(x)$: \[ y= y_i + (x-x_i) V_i \tan\alpha + \frac {g(x-x_i)^2}{2v_i^2 \cos^2\alpha} \]
\begin{remark}
Ballistics has been also 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.
\end{remark}