A car of mass M and a maximum power output of P is on an rough inclined plane Θ to the horizontal. What is the maximum velocity (v). Coefficient of friction=μ and air resistance=kv where k is constant

At the maximum velocity the driving force of the car is equal to the sum of the opposing forces: F_driving=F_friction+F_air+mgsinΘ F_friction=mgμcosΘ F_air=kv p=[mgμcosΘ+ kv+mgsinΘ]v = [μcosΘ+sinΘ]mgv+kv² kv²+[μcosΘ+sinΘ]mgv-p=0 solve using the quadratic equation: v= -[μcosΘ+sinΘ]mg ± [ ([μcosΘ+sinΘ]mg)²+4kp]^1/2 . 2k We only want the positive root as, the direction of velocity is up the incline therefore: v= -[μcosΘ+sinΘ]mg + [ ([μcosΘ+sinΘ]mg)²+4kp]^1/2 . 2k