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%%% Primary Title: ballistics (2D)
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This contributed entry contains basic contents of balistics as well as three live animations/videos of ballistics experiments to better illustrate this subject in \htmladdnormallink{classical mechanics}{http://planetphysics.us/encyclopedia/MathematicalFoundationsOfQuantumTheories.html}.
{\em Ballistics} is the study of the kinematics and \htmladdnormallink{dynamics}{http://planetphysics.us/encyclopedia/MathematicalFoundationsOfQuantumTheories.html} of a projected motion of an \htmladdnormallink{object}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}. Such an object in \htmladdnormallink{motion}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} 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
\htmladdnormallink{field}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} the latter also contributes to either accelerating or decelerating the
projectile depending on the initial conditions, i.e. the orientation and direction of an applied force relative to gravitational ones. The two following \htmladdnormallink{animations}{http://upload.wikimedia.org/wikipedia/commons/b/b6/Corioliskraftanimation.gif} provide interesting visual and dynamic examples of such \htmladdnormallink{projectile motions}{http://planetphysics.us/encyclopedia/ProjectileMotion.html} that are also represented graphically in the attached entry. As an interesting illustration of the problem, one can see that in the above linked animation the trajectories of a ball in the vertical and horizontal planes appear to be quite different; in the case of the vertical spinning disk the trajectory is simply a straight line, whereas on the horizontal spinning disk the trajectory of the same ball is quite visibly curved, but-- of course-- it can be seen again as a straight line trajectory in \htmladdnormallink{a rotating frame from above as shown in the linked web video}{http://video.google.com/videoplay?docid=-6469142924514101084}. In the next animation, also shown as a quick
\htmladdnormallink{movie download}{http://ww2010.atmos.uiuc.edu/(Gh)/guides/mtr/fw/gifs/coriolis.mpg}, one sees the
\htmladdnormallink{marked effects of a Coriolis force}{http://en.wikipedia.org/wiki/Coriolis_effect}, and more generally the effects of a spinning projectile (such as a football, soccer, tennis,ping-pong or golf ball) on its trajectory.
Similarly, a spinning \htmladdnormallink{black hole}{http://planetphysics.us/encyclopedia/BlackHoles.html} behaves quite differently from a non-spinning one, but one would not be able to correctly calculate such a major difference only with \htmladdnormallink{Newtonian mechanics}{http://planetphysics.us/encyclopedia/MathematicalFoundationsOfQuantumTheories.html}. Even \htmladdnormallink{special relativity}{http://planetphysics.us/encyclopedia/SR.html} is insufficient for this purpose, and General Relativity theory is needed.
The two-dimensional (2D) projection is a major step in defining the trajectory of a projectile object in its motion relative to that of an observer, or reference system. (Note also the
exceptions specified in the concluding remarks).
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 caused by the \htmladdnormallink{friction}{http://planetphysics.us/encyclopedia/Friction.html} with the atmosphere surrounding the object is negligible, the sum of forces along the $x$ axis $\Sigma F_x$ equals zero, and so the \htmladdnormallink{acceleration}{http://planetphysics.us/encyclopedia/Acceleration.html} along the axis $a_x$ is also insignificant. Thus, the \htmladdnormallink{velocity}{http://planetphysics.us/encyclopedia/Velocity.html} 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 \htmladdnormallink{position}{http://planetphysics.us/encyclopedia/Position.html}, $V_x$-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 \htmladdnormallink{mass}{http://planetphysics.us/encyclopedia/Mass.html} 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 \htmladdnormallink{function}{http://planetphysics.us/encyclopedia/Bijective.html} 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 \htmladdnormallink{types}{http://planetphysics.us/encyclopedia/Bijective.html} 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 ancient uses of various 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 \htmladdnormallink{programs}{http://planetphysics.us/encyclopedia/Program3.html} also make intensive use of ballistics, but in its more sophisticated forms; in such cases, the 2D projection method outlined above for ballistic calculations is obviously insufficient.
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