# Analytical Proof of Newton's Force Laws

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Analytical Proof of Newton s Force Laws Page 1 Analytical Proof of Newton s Force Laws 1 Introduction Many students intuitively assume that Newton s inertial and gravitational force laws F ma and Mm F G are true since they are clear and distance M m 2 simple However there is an analysis that ties the two equations together and demonstrates that they must be true The analysis provides answers to questions such as Is the inertial mass exactly the same as the gravitational mass Why is the exponent of distance 2 and not 1 99 or 2 01 or 1 Why is a constant required in one law and not in the other 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 and Kepler had determined the results So Newton in his 1669 paper Mathematical Principles of Philosophy now part of the Great Books Series in local libraries applied his force laws to the solar system and obtained the same results that Kepler had stated This confirmed Newton s ideas put physics on a firm mathematical basis and answered the above questions 2 Summary of Analytical Proof of Newton s Force Laws In the 8 step procedure that follows Newton s force laws are applied to the planet sun system and the planet earth path around the sun is shown to be an ellipse This procedure below uses the mathematics found in first year college texts and explains the mathematics within the derivations as they are being evolved 1 Observations show that the planets follow a smooth curve around the sun Sketch the planet at position P using polar coordinates r and within an orthogonal coordinate system Page 2 Analytical Proof of Newton s Force Laws 2 Differentiate planet position functions to obtain planet velocities v x and v y 3 Differentiate planet velocities to obtain planet accelerations ax and ay 4 Equate Newton s inertial and gravitational