Let be plane polar coordinates and and unit vectors in the direction of increasing and respectively. Show that the velocity of a particle moving in the plane with polar coordinates is given by
and that the unit normal to the particle path is parallel to
Deduce that the perpendicular distance from the origin to the tangent of the curve is given by
The particle, whose mass is , moves under the influence of a central force with potential . Use the conservation of energy and angular momentum to obtain the equation
Hence express as a function of as the integral
Evaluate the integral and describe the orbit when , with a positive constant.