The physical sciences seek to explain natural phenomena, and even to predict them. Simply observing these phenomena can inspire wonder. Explaining them brings satisfaction. Predicting them can be illuminating… or alarming. Another constant source of wonder is the remarkable fit between mathematics—which has a life of its own—and physics. This fruitful encounter between the two disciplines can take place through their curves and trajectories. How can the same curve describe the motions of objects as diverse as electrically charged particles and planets? How can a sine function capture the paths taken by light so well?
Particles, projectiles and planets -----------------------------------
In the physical sciences, conic sections describe all trajectories… or nearly all! Whether we are modelling a planet or comet orbiting a star, a projectile subject to little drag, or a charged particle approaching another particle or moving through a uniform electric field, the resulting trajectory is a conic section.
Any mechanical interaction that can be modelled by a central force may produce an elliptical, parabolic or hyperbolic trajectory. This term describes forces directed along the line through the centres of the two interacting objects O1 and O2, that is, forces of the form =f(r),\overset\to{F}=f(r)\overset\to{e_r}, where \overset\to{e_r} is a direction vector for the line (O1O2). In fact, \overset\to{e_r} is also a basis vector in the polar coordinate system attached to one of the objects under study (at point M) and centred on the other object with which it interacts (at O). Examples include the gravitational force between two masses and the electrostatic force between two electric charges.