拋體運動

拋體運動者，物理公式也，述拋一物之（二維）軌跡。其軌跡名拋物線也。

式

按運動方程，設$x$，$y$二軸；又設其始速$v_{0}$，交水平之角（自$x$軸逆時升之角）$\theta$，無氣阻則：

$v_{x}=v_{0}\cos \theta$

$v_{y}=v_{0}\sin \theta -gt$

$s_{x}=v_{0}\cos \theta t$

$s_{y}=v_{0}\sin \theta t-{\frac {1}{2}}gt^{2}$

然此未足。需計飛之時、高、水平距，兼系$x$，$y$座標之方程。下又設首尾垂直位移者 $d$。

一般之制

飛之時

$d=v_{0}\sin \theta t-{\frac {1}{2}}gt^{2}$

${\frac {1}{2}}gt^{2}-v_{0}\sin \theta t+d=0$

$t={\frac {v_{0}\sin \theta \pm {\sqrt {v_{0}^{2}\sin ^{2}\theta -2gd}}}{g}}$

然若$d<0$，則${\frac {v_{0}\sin \theta -{\sqrt {v_{0}^{2}\sin ^{2}\theta -2gd}}}{g}}$者負數也。故：

$t={\begin{cases}{\frac {v_{0}\sin \theta \pm {\sqrt {v_{0}^{2}\sin ^{2}\theta -2gd}}}{g}},h\geq 0\\{\frac {v_{0}\sin \theta +{\sqrt {v_{0}^{2}\sin ^{2}\theta -2gd}}}{g}},h<0\end{cases}}$

飛之高

$0=(v_{0}\sin \theta -g(0))^{2}-2gh$

$h={\frac {v_{0}^{2}\sin ^{2}\theta }{2g}}$

飛之水平距

$R$

$=v_{0}\cos \theta t$

$=v_{0}\cos \theta ({\frac {v_{0}\sin \theta \pm {\sqrt {v_{0}^{2}\sin ^{2}\theta -2gd}}}{g}})$

$={\begin{cases}{\frac {v_{0}^{2}\sin 2\theta \pm v_{0}{\sqrt {v_{0}^{2}\sin ^{2}2\theta -8gd\cos ^{2}\theta }}}{2g}},d\geq 0\\{\frac {v_{0}^{2}\sin 2\theta -v_{0}{\sqrt {v_{0}^{2}\sin ^{2}2\theta -8gd\cos ^{2}\theta }}}{2g}},d<0\end{cases}}$

座標方程

$s_{x}=v_{0}\cos \theta t$

$t={\frac {s_{x}}{v_{0}\cos \theta }}$

$s_{y}$

$=v_{0}\sin \theta ({\frac {s_{x}}{v_{0}\cos \theta }})-{\frac {1}{2}}g({\frac {s_{x}}{v_{0}\cos \theta }})^{2}$

$=s_{x}\tan \theta -{\frac {gs_{x}^{2}}{2v_{0}^{2}\cos ^{2}\theta }}$

即：

$y=x\tan \theta -{\frac {gx^{2}}{2v_{0}^{2}\cos ^{2}\theta }}$

$d=0$之殊況

若$d=0$，則：

$t$

$={\frac {v_{0}\sin \theta \pm {\sqrt {v_{0}^{2}\sin ^{2}\theta -2g(0)}}}{g}}$

$={\frac {2v_{0}\sin \theta }{g}}$

$R$

$={\frac {v_{0}^{2}\sin 2\theta \pm v_{0}{\sqrt {v_{0}^{2}\sin ^{2}2\theta -8g(0)\cos ^{2}\theta }}}{2g}}$

$={\frac {v_{0}^{2}\sin 2\theta }{g}}$