Some velocity and position calculations

Extract

Classically, the force of gravity between two masses m and M is given by the Principle of Universal Gravitation or the inverse-square law

where x is the distance between the centres of mass, and G is the universal gravitation constant (approximately 6.67x10^–11 m³ kg^–1s^–2). In elementary calculus, a common application is to assume that the acceleration due to gravity near the Earth’s surface is a constant, and then to do simple velocity and position calculations. If instead we apply the above law of force, then the acceleration due to gravity of an object of mass m is a function of the distance x = x(t) beween its centre of mass and the centre of the Earth at time t. Then, on applying F = ma

so the velocity and position calculations will be different. The purpose of this note is to present these calculations and their interesting consequences in a manner appropriate for a first year calculus course. (See also Temple B., and Tracey, C.A., “From Newton to Einstein”, American Mathematical Monthly, 99, 507–521,1992).