Ćwiczenia 27.04.09

mg  kv  =  m dv/dt 

$\frac{\text{dV}}{\text{mg} - \text{kv}}\ = \frac{1}{m}\text{\ dt}$ $- \frac{\frac{\text{dz}}{k}}{z}\text{\ \ } = - \frac{\text{dz}}{\text{kz}}\backslash t$

$$\int_{v_{0}}^{v}\frac{\text{dv}}{\text{mg} - \text{kv}}\ = \frac{1}{m}\ \ \int_{t_{0}}^{t}\text{dt}$$

$$P = \frac{t}{m}$$

$$L = (mg - kV = z\text{\ \ };\frac{\text{dz}}{\text{dv}}\ = - k;\ dV = \ \frac{- \text{dz}}{k})$$

$$L\ = \ \int_{}^{}{- \frac{\frac{\text{dz}}{k}}{z}}\ \ = \ - \frac{1}{k}\ \ \int_{}^{}\frac{\text{dz}}{z}\ = \ - \frac{1}{k}\ \ln\left( mg - \text{kV} \right)\ |\begin{matrix} v \\ v_{0} \\ \end{matrix}\ = - \frac{1}{k}\ln\left( ,g - kV \right) + \frac{1}{k}\ln\left( mg - kV_{0\ } \right) = \ - \frac{1}{k}ln(\frac{mg - kV}{mg - kV_{0}})$$

$$\ln\frac{mg - kV}{mg - kVo} = - \frac{k}{m}\text{\ t}$$

$\frac{mg - kV}{mg - kVo}\ = \text{\ e}^{- \frac{k}{m}t}\ $

$$mg - kV\ = \text{\ e}^{- \frac{k}{m}t}(mg - kV_{0})\ $$

$$kV\ = {\text{mg} - \text{\ e}}^{- \frac{k}{m}t}(mg - kV_{0})$$

$$V\ = {\text{mg} - (mg - kV_{0}) - \text{\ e}}^{- \frac{k}{m}t}$$

$$V\ = {\frac{\text{mg}}{k} - (\frac{\text{mg}}{k} - V_{0}) - \text{\ e}}^{- \frac{k}{m}t}$$

mg  =  k v