MOTION, EQUATION OF

mathematical formula that describes the position, velocity, or acceleration of a body relative to a given frame of reference. Newton's second law, which states that the force F acting on a body is equal to the mass m of the body multiplied by the acceleration a of its centre of mass, F = ma, is the basic equation of motion in classical mechanics. If the force acting on a body is known as a function of time, the velocity and position of the body as functions of time can, theoretically, be derived from Newton's equation by a process known as integration. The gravitational force (g) acting on a falling body, for example, accelerates it and is the weight of the body W, which is constant with respect to time; thus, using Newton's equation, W = mg and m = W/g. Substituting these values in Newton's equation, W = (W/g)a, from which a = g, the acceleration of gravity. Acceleration is the rate of change of velocity with respect to time, so that by integration the velocity v in terms of time t is given by v = gt. Velocity is the time rate of change of position S, and, consequently, integration of the velocity equation yields S = 1/2gt2. If the force acting on a body is specified as a function of position or velocity, the integration of Newton's equation may be more difficult. When a body is constrained to move in a specified manner on a fixed path, it may be possible to derive the position-time equation; from this equation the velocity-time and acceleration-time equations can, theoretically, be obtained by a process known as differentiation.