Newton's Second Law

Statement. The force acting on an object equals the rate of change of its momentum:

If the object's mass $m$ stays constant during the motion (the usual case), this simplifies:

This is a genuinely vector equation: force and the resulting acceleration always point in exactly the same direction, and in Cartesian components it splits into three independent scalar equations, $F_x=ma_x$, $F_y=ma_y$, $F_z=ma_z$ — a force along one axis changes only the acceleration along that same axis and has no effect on the other two.

Unit of force. The SI unit, the newton (N), is defined directly from this law: one newton is the force which, acting on a mass of 1 kg, produces an acceleration of $1\text{ m s}^{-2}$ in the direction of the force.

Key features of the second law:

• It relates force and acceleration at the same instant — the acceleration at time $t$ depends only on the force acting at time $t$, never on any force that acted earlier.
• If several forces $\vec F_1,\vec F_2,\ldots,\vec F_n$ act simultaneously, the net force determining the acceleration is their vector sum, $\vec F_{net}=\vec F_1+\vec F_2+\cdots+\vec F_n$.
• Since $\vec a=\dfrac{d^2\vec r}{dt^2}$, the law can equivalently be written $\vec F=m\dfrac{d^2\vec r}{dt^2}$, a second-order differential equation for position. …