The inverse velocity force and automotive physics

The inverse velocity force F(v)=C/v is central to the analysis of automotive performance, but is often unmentioned in classical mechanics courses. It represents the maximum force that can be delivered from a power-limited engine over a wide range of speeds. We show how such a force arises from a device called a transmission in internal combustion engines, and more simply from circuit controls in electric vehicles. We also examine how the force needs to be modified and supplemented to describe vehicle acceleration throughout the entire range of speeds, along with the ensuing kinematics. Along the way we compare gasoline vehicles with electric ones and derive the shape of their torque-speed curves.