Equations of motion are used to determine the velocity, displacement or acceleration of an object in constant motion. Most applications of the equations of motion are used to express how an object moves under the influence of a constant, linear force. Variations of the basic equation are used to account for objects moving on a circular path or in a pendulum configuration.

An equation of motion, also referred to as a differential equation of motion, mathematically and physically relates Newton’s second law of motion. The second law of motion, according to Newton, states that a mass under the influence of a force will accelerate in the same direction as the force. Force and magnitude are directly proportional, and force and mass are inversely proportional.

Standard equations of motion involve five variables. One variable is for the starting and ending position of the object, also known as displacement. Two variables represent the initial and final velocity measurements, respectively known as the change in velocity. The fourth variable describes acceleration. The fifth variable stands for the time interval.

The classic equation to solve the linear acceleration of an object is written as the change in velocity divided by the change in time. The law of motion equation typically is set up using three kinetic variables: velocity, displacement and acceleration. Acceleration can be solved for by using velocity and displacement as long as the second law of motion applies to the problem.

When an object is in constant acceleration along a rotational trajectory, the equations of motion are different. In this situation, the classic equation for circular acceleration of an object is written using the initial and angular velocities, angular displacement and angular acceleration.

A more complicated application of the equations of motion is the pendulum equation of motion. The basic equation is known as Mathieu’s equation. It is expressed using the gravity constant for acceleration, the length of the pendulum and the angular displacement.

There are several assumptions that must be satisfied to use such an equation for a problem involving a pendulum configuration. The first assumption is that the rod that connects the mass to the axis point is weightless and remains taut. The second assumption is the motion is limited to two directions, back and forth. The third assumption is that the energy lost to air resistance or friction is negligible. Variations of the basic equation are used to account for infinitesimal oscillations, compound pendulums and other configurations.