Newton’s Laws of Motion

Why It Matters

Newton’s laws are the foundation of dynamics. They explain how forces relate to motion, why a force is needed to change velocity rather than maintain velocity, and how forces between interacting bodies occur in pairs.

Definition

Newton’s first law states that a body remains at rest or moves with constant velocity unless acted on by a non-zero resultant external force.

Newton’s second law states that the resultant force on a body equals the rate of change of momentum:

$${\vec{F}}_{\text{net}}=\frac{d\vec{p}}{dt}$$

For constant mass, this becomes:

$${\vec{F}}_{\text{net}}=m\vec{a}$$

Newton’s third law states that if body A exerts a force on body B, then body B exerts an equal and opposite force on body A. These two forces act on different bodies.

Key Representations

For a body of constant mass:

$$\sum \vec{F}=m\vec{a}$$

In component form:

$$\sum F_x=m{a}_x$$ $$\sum F_y=m{a}_y$$

These are signed component equations after choosing positive $x$- and $y$-directions. In one dimension, the vector equation is often written as one signed scalar equation along the chosen positive direction.

If the resultant force is zero:

$$\sum \vec{F}=\vec{0}$$

then:

$$\vec{a}=\vec{0}$$

The body may be at rest or moving with constant velocity.

Common Exam Points

• Resultant force causes acceleration, not velocity.
• A body can move at constant velocity with zero resultant force.
• Free-body diagrams show forces acting on one chosen body only.
• Third-law pairs act on different bodies, so they do not cancel on the same free-body diagram.
• Weight is a force: $\vec{W}=m\vec{g}$.
• Normal contact force is not automatically equal to weight; it depends on the acceleration and other vertical forces.

Links

• Prerequisite: kinematics
• Prerequisite: forces
• Prerequisite: vectors
• Related: dynamics
• Related: force diagrams and resolution
• Related: newtonian dynamics applications
• Related: momentum and impulse
• Misconception: newtons laws distinction
• Misconception: force identification errors
• Misconception: vector direction consistency