Cinematica del punto

$$\overrightarrow{v_P\ }=\ \frac{\text{d}x_P}{\text{d}t}\overrightarrow{i}+\frac{\text{d}y_P}{\text{d}t}\overrightarrow{j}+\frac{\text{d}z_P}{\text{d}t}\overrightarrow{k}$$  

$$\overrightarrow{v_P\ }=\frac{\text{d}s}{\text{d}t}\overrightarrow{t}$$  

$$\overrightarrow{v_P\ }=\ \sqrt{\left(\frac{\text{d}x_P}{\text{d}t}\right)^{2\ }+\left(\frac{\text{d}y_P}{\text{d}t}\right)^{2\ }+\left(\frac{\text{d}z_P}{\text{d}t}\right)^{2\ }}\ \ \overrightarrow{t}$$  

$$\overline{v_P\ }\ =\ \frac{\text{d}\rho}{\text{d}t}e^{i\theta}+\rho\ \frac{\text{d}\theta}{\text{d}t}e^{i\left(\theta+\frac{\pi}{2}\right)}$$  

 $\overrightarrow{a_P\ }\ =\ \frac{\text{d}^2s}{\text{d}t^2}\overrightarrow{n\ }+\frac{1}{\rho}\left(\frac{\text{d}s}{\text{d}t}\right)^2\ \overrightarrow{t}$  

 $\overrightarrow{a_P\ }\ =\ \frac{\text{d}^2s}{\text{d}t^2}\overrightarrow{t\ }+\frac{1}{\rho}\left(\frac{\text{d}s}{\text{d}t}\right)^2\ \overrightarrow{n}$  

 $\overrightarrow{a_P\ }\ =\ \frac{\text{d}^2s}{\text{d}t^2}\overrightarrow{n\ }+\left(\frac{\text{d}s}{\text{d}t}\right)^2\ \overrightarrow{t}$  

 $\overrightarrow{a_P\ }\ =\ \frac{\text{d}^2s}{\text{d}t^2}\overrightarrow{t\ }$  

 $\overrightarrow{a_P\ }\ =\ \frac{\text{d}^2\rho}{\text{d}t^2}e^{i\theta}+\rho\ \left(\frac{\text{d}\theta}{\text{d}t}\right)^2\ e^{i\left(\theta+\pi\right)}$  

 $\overrightarrow{a_P\ }\ =\ \sqrt{\left(\frac{\text{d}^2x_P}{\text{d}t^2}\right)^2+\left(\frac{\text{d}^2y_P}{\text{d}t^2}\right)^2}\overrightarrow{t}$  

 $\overrightarrow{a_P}\ =\ a_{P^{ }}^t\overrightarrow{t}+a_{P^{ }}^n\overrightarrow{n}$  

 $\overrightarrow{a_P}\ =\ \frac{\text{d}v_P}{\text{d}t}\overrightarrow{t}\ +\frac{v_P^2}{\rho}\overrightarrow{n}$  

$\overrightarrow{t}\ =\ \frac{\overrightarrow{v_P}}{\left|\overrightarrow{v_P}\right|}$

$\overrightarrow{n}\ =\ \overrightarrow{b}\ \times\overrightarrow{t}$

$\overrightarrow{b}\ =\ \overrightarrow{t}\times\overrightarrow{n}$

$\overrightarrow{b}\ =\ \overrightarrow{n}\times\overrightarrow{t}$

$\overrightarrow{a_P^n\ }=\ 0$

$\overrightarrow{t}\ =\ \overrightarrow{i}$

$\overrightarrow{t}\ =\ \cos\left(\alpha\right)\overrightarrow{i}+\sin\left(\alpha\right)\overrightarrow{j}$

$\overrightarrow{a_P}\ =\ \left(\frac{\text{d}s}{\text{d}t}\right)^2\overrightarrow{t}$

$s\ =\ R\omega t$

$\overrightarrow{v_P\ }\ =\ \rho\frac{\text{d}\theta}{\text{d}t}e^{i\theta}$

$\overrightarrow{v_P}\ =\ R\omega$

$\overrightarrow{v_P}\ =\ \frac{\text{d}\rho}{\text{d}t}e^{i\theta}$

$\overrightarrow{a_P}\ =\ 0$

$\overrightarrow{a_P}\ =\ \omega R\overrightarrow{t}+\omega^2R\overrightarrow{n}$

$\overrightarrow{a_P}\ =\ \omega^2R\overrightarrow{n}$

$\overrightarrow{a_P}\ =\ \omega R\overrightarrow{t}$

$\left|\overrightarrow{a_P}\right|\ =\ \sqrt{\left(\rho\frac{\text{d}^2\theta}{\text{d}t^2}\right)^2+\left(\rho\left(\frac{\text{d}\theta}{\text{d}t}\right)^2\right)^2}$

$\overrightarrow{a_p}^t\ =\rho\frac{\text{d}^2\theta}{\text{d}t^2}e^{i\left(\theta+\frac{\pi}{2}\right)}\$

$\overrightarrow{a_p}^n\ =-\rho\left(\frac{\text{d}\theta}{\text{d}t}\right)^2e^{i\theta}\$

$\overrightarrow{a_p}^n\ =-\rho\left(\frac{\text{d}\theta}{\text{d}t}\right)^2e^{i\left(\theta+\pi\right)}$

$\left|\overrightarrow{a_P}\right|\ \ =\ \sqrt{\left(\frac{\text{d}^2s}{\text{d}t^2}\right)^2+\left(\frac{\text{d}s}{\text{d}t}\right)^4}$

$\left|\overrightarrow{a_P}\right|\ =\ \frac{\text{d}^2x_P}{\text{d}t^2}+\frac{\text{d}^2y_P}{\text{d}t^2}$

$\left|\overrightarrow{a_P}\right|\ \ =\ \sqrt{\left(\rho\frac{\text{d}^2\theta}{\text{d}t^2}\right)^2+\left(\rho\left(\frac{\text{d}\theta}{\text{d}t}\right)^2\right)^2}$

$\left|\overrightarrow{a_P}\right|\ \ =\ R\frac{\text{d}^2\theta}{\text{d}t^2}$

$$\overrightarrow{t}=\ \frac{A\overrightarrow{i}-2Bt\overrightarrow{j}}{\sqrt{A^2+4B^2t^2}}$$

$$\left|\overrightarrow{a_P}\right|\ =\ 2B$$

$$y\ =\ \frac{B}{A^2}x^2$$

$$\overrightarrow{v_P}\ =\ \sqrt{\left(At\right)^2+\left(-Bt^2\right)^2\ }\overrightarrow{t}$$