Newton's Third Law Calculator

Newton's third law states that when one object exerts a force on another, the second object simultaneously exerts an equal-magnitude force in the opposite direction on the first. These forces are commonly called an action-reaction pair. They arise from one interaction, occur at the same time, and act on different objects. This calculator displays the signed pair and combines each force with the corresponding object's mass to illustrate the accelerations produced by that interaction. Suppose object A pushes object B with a positive force. Object B receives that positive action force, while object A receives an equal negative reaction force. Applying Newton's second law separately gives the acceleration of A as the negative reaction force divided by mass A and acceleration of B as the positive action force divided by mass B under this calculator's sign convention. The displayed acceleration directions are opposite. If the masses differ, acceleration magnitudes differ even though the forces are equal: the lighter object changes velocity more rapidly. Action and reaction do not cancel each other on either individual object because they act on separate bodies. Forces cancel only when equal and opposite forces act on the same chosen system. If A and B are analyzed together as one system, their internal interaction forces do cancel in the system-wide force balance. External forces can still accelerate the combined center of mass. This distinction is essential when drawing free-body diagrams. Walking offers a familiar example: a foot pushes the ground backward and the ground pushes the walker forward. A rocket pushes exhaust backward while exhaust pushes the rocket forward. A book presses downward on a table while the table experiences that force; the table pushes upward on the book as the paired contact force. The upward normal force on the book and its downward weight may balance, but those two forces are not a third-law pair because both act on the same object and come from different interactions. The calculator isolates one mutual force and assumes each object can be treated as a particle. Other external forces such as friction, gravity, drag, or support reactions must be included separately to obtain each object's net acceleration. In an isolated two-object system, momentum changes are equal and opposite, preserving total momentum. Use this model to understand signs, compare accelerations, and check introductory mechanics problems.