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How does classical mechanics explain the motion of objects under forces?

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A diagram with labeled vector illustrations. Panel A shows vector addition with two green arrows A and B forming a parallelogram, and their sum C as a red diagonal. Panel B shows vector subtraction, with A and negative B forming resultant D. Panel C demonstrates vector components using angles and projections. Panel D uses the right-hand rule with thumbs to show the direction of the cross product E = A x B and its opposite, -E = B x A. — mechanics Figure 1: (A) The vector sum C = A + B = B + A. (B) The vector difference A + (−B) = A − B = D. (C, left) A cos θ is the component of A along B and (right) B cos θ is the component of B along A. (D, left) The right-hand rule used to find the direction of E = A × B and (right) the right-hand rule used to find the direction of −E = B × A.

Classical mechanics explains the motion of objects under forces through the three laws of motion formulated by Isaac Newton in his Principia. These laws provide a framework for predicting how objects behave when subjected to external influences:312

First Law (Law of Inertia): An object remains at rest or in uniform motion in a straight line unless acted upon by an unbalanced force.31
Second Law: The time rate of change of an object's momentum is proportional to the applied force and occurs in the same direction. For an object with constant mass, this is expressed as $𝐅 = m 𝐚$ , where $𝐅$ is the net force, $m$ is mass, and $𝐚$ is acceleration.31
Third Law (Action and Reaction): For every action, there is an equal and opposite reaction. When two bodies interact, they exert forces on each other that are equal in magnitude and opposite in direction.31

If the net force on an object is zero, it remains in equilibrium; if a net force exists, the object accelerates.31