3.4 Trajectory of a projectile

  Motion in Two Dimension

Consider the body undergoing projectile motion be at the point $P$ at any time $t$ as shown in the figure.

Let the co=ordinate of point $P$ be $(x,y)$

$x:$ Displacement of body along $x-$ axis in time $t$
$y:$ Displacement of body along $y-$ axis in time $t$

Therefore, kinematics equation with relates displacement and time $t$ is,
$${\overrightarrow s _x} = {\overrightarrow u _x}t + \frac{1}{2}{\overrightarrow a _x}{t^2}$$$$x = \left( {u\cos \theta } \right)t + 0$$$$x = \left( {u\cos \theta } \right)t$$or$$t = \frac{x}{{u\cos \theta }} \quad ...(i)$$
Similarly, $${\overrightarrow s _y} = {\overrightarrow u _y}t + \frac{1}{2}{\overrightarrow a _y}{t^2}$$$$y = \left( {u\sin \theta } \right)t + \frac{1}{2}\left( { - g} \right){t^2}$$$$y = \left( {u\sin \theta } \right)t - \frac{1}{2}g{t^2}\quad ...(ii)$$ From equation $(i)$ and $(ii)$ we get,
$$y = \left( {u\sin \theta } \right)\left( {\frac{x}{{u\cos \theta }}} \right) - \frac{1}{2}g{\left( {\frac{x}{{u\cos \theta }}} \right)^2}$$$$y = x\tan \theta - \frac{{g{x^2}}}{{2{u^2}{{\cos }^2}\theta }}$$$$y = x\tan \theta - \frac{{g{x^2}}}{{2{u^2}}}{\sec ^2}\theta $$

The above equation is the trajectory equation of projecticle motion and is same as that of parabola.