However, there is an analysis that ties the two equations together and demonstrates that they must be true. The ideal way to prove new theoretical laws is to forecast the outcome of an experiment using the laws, perform the experiment, and find that the result is as forecast. But nature had already performed the experiment with planets in the solar system, a nd Kepler had determined the results. This confirmed Newton's ideas, put physics on a firm mathematical basis and answered the above questions. This procedure below uses the mathematics found in first year college texts and explains the mathematics within the derivations as they are being evolved.
Observations show that the planets follow a smooth curve around the sun.
Sketch the planet at position P , using polar coordinates, r and q , within an orthogonal coordinate system Differentiate planet position functions to obtain planet velocities, . Differentiate planet velocities to obtain planet accelerations, . Equate Newton's inertial and gravitational force laws as applied to the planet. In this step we assume that inertial mass is identical to gravitational mass and that the force of gravity decreases as the square of distance. G must also be assumed to make the equations consistent.