# Linear motion equation with intital velocity and water drag

at the start i want to mention that i am from germany, so my english is not the best!

For a boat simultaion project i am working linear motion equation with intital velocity and water drag.

For simplicity i use drag with: $$bv^2$$ -> For a Boat with motorforce F i got this equation $$ma + bv^2 = F$$ I solved it by using $$v_{max} = \sqrt{F/b}$$ and got $$v = v_{max} * \tanh(\sqrt{Fb}/m * t + c)$$ For inital velocity -> c = 0 the equation graph looks fine. But wenn i set c to something higher then v_max the graph is just moved to the left. But i need an equation where you can put your intial velocity in and a maximal velocity (depending on the motorforce) and the velocity starts to rise or fall after t = 0!

$$v= v_{init} - (v_{init} - v_{max})*tanh(\sqrt{Fb}/m * t)$$ $$s= m(\sqrt{F/b} - v_{init}) * ln(cosh(t\sqrt{F*b}/m))/\sqrt{F*b)} + v_{init}t + s_0$$
$$v(t) = v_{max}\tanh\left(\frac{\sqrt{bF}}{m}t+\tanh^{-1}\frac{v_0}{v_{max}}\right)$$
where $v_0$ is the initial speed.