Variable resistive forces

Most of the time resistive forces change with speeds

Example

Given the following system, when the vehicles acceleration is $0.8\text{ ms}^{-2}$, its engine is working at $30\text{ kW}$. Find its velocity.

$$\begin{align} F &= ma\\ F - 80v - 1800g\sin(10^{\circ}) &= 1800 \times 0.8\\ F &= 4503.154 + 80v \end{align}$$ $$\begin{align} P &= Fv\\ 30000 &= v(4503.154 + 80v)\\ &= 80v^2 + 4503.154v\\ 0 &= 80v^2 + 4503.154v - 30000\\ 0 &= v^2 + 56.3v - 375\\ v &= 6.02\text{ ms}^{-1}, -62.3\text{ ms}^{-1} \end{align}$$

Since the vehicles is moving up the slope $v = 6.02\text{ ms}^{-1}$