Projectile motion is the motion of an object which is being influenced by gravity after being launched. The values which do not change for a projectile are:
- $u_x=u\cos{\theta}$
- $u_y=u\sin{\theta}$
- $a_x=0$ (no horizontal forces)
- $a_y=-g$ (the force of gravity acts vertically downwards, downwards being negative)
Applying the horizontal direction to the equation, $v=u+at$: $$ \begin{equation}\begin{aligned} v&=u+at\\ v_x&=u_x+a_xt\\ v_x&=u\cos{\theta}+(0)t\\ v_x&=u\cos{\theta}\\ v_x&=u_x\\ \end{aligned}\end{equation} $$
Applying the vertical direction to the equation, $v=u+at$: $$ \begin{equation}\begin{aligned} v&=u+at\\ v_y&=u_y+a_yt\\ v_y&=u\sin{\theta}+(-g)t\\ v_y&=u\sin{\theta}-gt\\ \end{aligned}\end{equation} $$
Applying the horizontal direction to the equation, $s=ut+\frac12at^2$: $$ \begin{equation}\begin{aligned} s&=ut+\frac12at^2\\ s_x&=u_xt+\frac12a_xt^2\\ s_x&=(u\cos{\theta})t+\frac12(0)t^2\\ s_x&=ut\cos{\theta}\\ \end{aligned}\end{equation} $$
Applying the vertical direction to the equation, $s=ut+\frac12at^2$: $$ \begin{equation}\begin{aligned} s&=ut+\frac12at^2\\ s_y&=u_yt+\frac12a_yt^2\\ s_y&=(u\sin{\theta})t+\frac12(-g)t^2\\ s_y&=ut\sin{\theta}-\frac12gt^2\\ \end{aligned}\end{equation} $$
Applying the horizontal direction to the equation, $v^2=u^2+2as$: $$ \begin{equation}\begin{aligned} v^2&=u^2+2as\\ v_x^2&=u_x^2+2a_xs_x\\ v_x^2&=(u\cos{\theta})^2+2(0)s_x\\ v_x^2&=u^2\cos^2{\theta}\\ \end{aligned}\end{equation} $$
Applying the vertical direction to the equation, $v^2=u^2+2as$: $$ \begin{equation}\begin{aligned} v^2&=u^2+2as\\ v_y^2&=u_y^2+2a_ys_y\\ v_y^2&=(u\sin{\theta})^2+2(-g)s_y\\ v_y^2&=u^2\sin^2{\theta}-2gs_y\\ \end{aligned}\end{equation} $$
Special formulae
The special formulae are time of flight, horizontal range and maximum height: $$ \begin{equation}\begin{aligned} \text{time of flight}&=\frac{2u\sin{\theta}}{g}\\ \text{horizontal range}&=\frac{u^2\sin{2\theta}}{g}\\ \text{max height}&=\frac{u^2\sin^2{\theta}}{2g}\\ \end{aligned}\end{equation} $$