Differential equations

If we have a varible acceleration caused by an overall force we can use differential equatiosn to describe the bodys motion: $F = m\frac{dv}{dt}$.

Example

At time $t = 0$ seconds, a mass of 5 kg begins to fall downwards. Their is a restive force of 4v N. Find v in terms of t.

Use newtons second law

$$\begin{align} F &= ma\\ 5g - 4v &= 5 \times \frac{dv}{dt}\\ g - \frac{4}{5}v &= \frac{dv}{dt}\\ \frac{1}{g - \frac{4}{5}v} &= \frac{dt}{dv}\\ \int{\frac{1}{g - \frac{4}{5}v} \text{ } dv} &= \int{\frac{dt}{dv} \text{ } dv}\\ -\frac{5}{4}\ln\left|g - \frac{4}{5}v\right| + c &= \int{dt}\\ -\frac{5}{4}\ln\left|g - \frac{4}{5}v\right| &= t + C\\ \end{align}$$

Solve the constant of integration

When $t = 0$, $v = 0$ so: $$\begin{align} -\frac{5}{4}\ln|g| &= C\\ \end{align}$$

Rearrange

$$\begin{align} -\frac{5}{4}\ln\left|g - \frac{4}{5}v\right| &= t -\frac{5}{4}\ln|g|\\ \ln\left|g - \frac{4}{5}v\right| &= -\frac{4}{5}t + \ln|g|\\ g - \frac{4}{5}v &= e^{-\frac{4}{5}t + \ln|g|}\\ g - \frac{4}{5}v &= e^{-\frac{4}{5}t} \times e^{\ln|g|}\\ g - \frac{4}{5}v &= ge^{-\frac{4}{5}t}\\ -\frac{4}{5}v &= ge^{-\frac{4}{5}t} - g\\ v &= -\frac{5}{4}g(e^{-\frac{4}{5}t} - 1)\\ \end{align}$$