12.9: Summary

This following table summarizes the formulæ for a projectile initially at the origin, fired with initial velocity $v_{0}$ at an angle $\theta$ from the horizontal.

Table 9-1. Summary of formulæ for projectile motion.

Quantity	Formula
$x(t)$	$x=\left(v_{0} \cos \theta\right) t$
$y(t)$	$y=-\dfrac{1}{2} g t^2+\left(v_0 \sin \theta\right) t$
$y(x)$	$y(x)=\left(-\dfrac{g}{2 v_{0}^{2} \cos ^{2} \theta}\right) x^{2}+(\tan \theta) x$
Time in flight	$t_{f}=\dfrac{2}{g} v_{0} \sin \theta$
Range at angle $\theta$	$R=\dfrac{v_{0}^{2}}{g} \sin 2 \theta$
Max. range (at $\left.\theta=45^{\circ}\right)$	$R_{\max }=\dfrac{v_{0}^{2}}{g}$
Angle needed to hit target at range $R$ for fixed $v_{0}$	$\theta=\dfrac{1}{2} \sin ^{-1}\left(\dfrac{g R}{v_{0}^{2}}\right)$
Speed needed to hit target at range $R$ for fixed $\theta$	$v_{0}=\sqrt{\dfrac{g R}{\sin 2 \theta}}$
Max. altitude	$h=\dfrac{v_{0}^{2} \sin ^{2} \theta}{2 g}$
Speed needed to hit target at $\left(x_{t}, y_{t}\right)$ for fixed $\theta$	$\left(x_{t}, y_{t}\right)$ for fixed $\theta$ & $v_{0}=\sqrt{\dfrac{g x_{t}}{2\left(\tan \theta-\dfrac{y_{t}}{x_{t}}\right) \cos ^{2} \theta}}$
Angle needed to hit target at $\left(x_{t}, y_{t}\right)$ for fixed $v_{0}$	$x_{t} \sin 2 \theta-2 y_{t} \cos ^{2} \theta=\dfrac{g x_{t}^{2}}{v_{0}^{2}}$

Note that this last expression must be solved iteratively for $\theta$.