# Problems
###### tags: `Physics`
### Chap 5. Particle Dynamics

$$
\left.\begin{array}{r}a=\frac Fm\Rightarrow\frac{d^2y}{dt^2}=\frac{mg({\displaystyle\frac yL})}m=g\frac yL\\\\a=\frac{dv}{dt}=\frac{dv}{dy}\cdot\frac{dy}{dt}=\frac{dv}{dy}\cdot v\end{array}\right\}\Rightarrow vdv=\frac gLydy\Rightarrow\int_0^vvdv=\int_{y_0}^L\frac gLydy\\\Rightarrow v^2=\frac gL(L^2-y_0^2)\Rightarrow v=\sqrt{\frac gL(L^2-y_0^2)}
$$
### Chap 7. Work and Energy
$$
P=\frac{dE_k}{dt}=\frac{{\displaystyle\frac12}mdv^2}{dt}\\\Rightarrow\frac12m\int dv^2=\int Pdt\Rightarrow\frac12mv^2=Pt\Rightarrow v=\frac{dx}{dt}=\sqrt{\frac{2Pt}m}\\\int dx=\int\sqrt{\frac{2Pt}m}dt\Rightarrow
$$