运动学

基本公式

$\begin{array}{|c|c|} \hline v_{0},v,a, t & v=v_{0}+a t \\ \hline v_{0}, a, t, x & x=v_{0} t+\frac{1}{2} a t^{2} \\ \hline v_{0}, v, a, x & v^{2}-v_{0}^{2}=2 a x \\ \hline v_{0}, v, t, x & \color{red}{x=\frac{v+v_{0}}{2} t} \\ \hline \end{array}$

推论

$\bar{v}=\frac{v_{0}+v}{2}=v_{\frac{t}{2}}$ $v_{\frac{x}{2}}=\sqrt{\frac{v_{0}^{2}+v^{2}}{2}}$ $\triangle x=x_n-x_{n-1}=at^2$ $x_{1}: x_{2}: x_{3}: \cdots: x_{n}=1^{2}: 2^{2}:3^{2}: \cdots: n^{2}$ $x_{\mathrm{I}}: x_{\mathrm{II}}: x_{\mathrm{III}} : \cdots: x_{N}=1: 3: 5: \cdots:(2 n-1)$ $t_{1}: t_{2}: \cdots: t_{n}=1:(\sqrt{2}-1):(\sqrt{3}-\sqrt{2}): \cdots:(\sqrt{n}-\sqrt{n-1})$ $a=\frac{(x_6+x_5+x_4)-(x_3+x_2+x_1)}{9T^2}=\frac{\left(\frac{d}{\triangle t_2}\right)^2-\left(\frac{d}{\triangle t_1}\right)^2}{2s}$

平抛运动

$\begin{cases} x=v_0t\\ y=\frac12gt^2 \end{cases}\Rightarrow y=\frac g{2v_0^2}x^2$ $\tan\varphi=2\tan\theta=\frac{gt}{v_0}$

斜抛运动

$\begin{cases} x=v_0\cos \theta t\\ y=v_0\sin \theta t-\frac12gt^2 \end{cases} \Rightarrow y=-\frac g{2v_0\cos\theta}x^2+\tan\theta x$

圆周运动

$\begin{aligned}a&=\omega v=\omega^2r=\frac{v^2}r\\&=\left(\frac{2\pi}T\right)^2r=(2\pi f)^2r\\&=(2\pi n)^2r\end{aligned}$ $\begin{cases} v=s/t\rightarrow \text{m/s}\\ \omega=\theta /t\rightarrow \text{rad/s}\\ T=2\pi/\omega\rightarrow \text{s}\\ f=1/T \rightarrow\text{r/s}\\ n=1/T \rightarrow\text{Hz} \end{cases}$