equation of motion

Velocity-time curves(Top left) Velocity-time curve for uniform motion, that is, an object moving at a constant velocity. The area under the curve defined by the rectangle ABDC is the distance the object moved between times t1 and t2. (Top right) Velocity-time curve for uniform acceleration. The velocity increases at a linear rate, and as in the graph for uniform motion, the area under the curve between A and E is the distance the object moved between times t1 and t2. (Bottom) Two examples of nonuniformly accelerated motion. (a) Although the slope is that of uniform deceleration, the points are scattered above and below the line. (b) The object undergoes periods of accelerated motion (positive slope) and decelerated motion (negative slope).

equation of motion, 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 center 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.

For example, a falling body accelerates at a constant rate, g. 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. If one graphs acceleration versus time, the velocity v would be the area under the curve. Velocity is the time rate of change of position S, and, consequently, integration of the velocity equation yields S = 1/2gt2. Similarly, if one graphs velocity versus time, the area under the curve gives the change in position, and the derivative of the velocity, or the slope of the velocity curve would be the acceleration (in this case, g). If a body is moving with a uniform velocity, the slope of the velocity curve would be zero.

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 differentiation.

The Editors of Encyclopaedia BritannicaThis article was most recently revised and updated by Erik Gregersen.
Citation Information
Article Title: equation of motion
Website Name: Encyclopaedia Britannica
Publisher: Encyclopaedia Britannica, Inc.
Date Published: 12 September 2026
Access Date: November 14, 2026