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: Fdriving=Ffriction+Fair+mgsinΘ Ffriction=mgμcosΘ Fair=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
