Ballistics (2D)Ballistics2006-07-25 16:23:332009-03-06 17:53:18Definition% this is the default PlanetPhysics preamble. as your knowledge
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% define commands hereBallistics 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:
$$\[y=y_i + (x-x_i) V_i \tan\alpha + \frac {g(x-x_i)^2}{2{v_i}^2 \cos^2\alpha}\]$$
\subsection{2D Motion:}
The merging of the functions of place-time $x(t),y(t)$ produces the route equation $y(x)$: \[ y= y_i + (x-x_i) V_i \tan\alpha + \frac {g(x-x_i)^2}{2v_i^2 \cos^2\alpha} \]