A motorcycle of mass moves on a bowl-shaped surface specified by its height where is the radius in cylindrical polar coordinates . The torque exerted by the motorcycle engine on the rear wheel results in a force pushing the motorcycle forward. Assuming is directed along the motorcycle's velocity and that the motorcycle's vertical velocity and acceleration are small, show that the motion is described by
where dots denote time derivatives, and is the acceleration due to gravity.
The motorcycle rider can adjust to produce the desired trajectory. If the rider wants to move on a curve , show that must obey
Now assume that , with a constant, and with a positive constant, and so that the desired trajectory is a spiral curve. Assuming that tends to infinity as tends to infinity, show that tends to and tends to as tends to infinity.