Momentum and Newton's Second Law of Motion

Before stating Newton's second law, it helps to introduce the quantity it is actually written in terms of: linear momentum. The momentum of a body of mass $m$ moving with velocity $v$ is defined as

Momentum is a vector quantity, pointing in the same direction as the body's velocity, and its SI unit is $\text{kg}\,\text{m/s}$ (equivalently, $\text{N}\,\text{s}$, as will become clear once impulse is introduced in the next section). Momentum captures something that velocity alone does not: a heavy truck moving slowly and a small car moving fast can, for suitable masses and speeds, carry the very same momentum, and both would require the same overall push, sustained over the same time, to bring them to rest -- which is precisely why momentum, not velocity by itself, is called a body's "quantity of motion."

Newton's second law of motion states that the rate of change of a body's momentum is directly proportional to the net external force applied to it, and this change takes place in the direction of the applied force:

The SI unit of force, the newton (N), is defined precisely so that the constant of proportionality $k$ equals $1$, giving the working form of the second law:

For a body of constant mass $m$, since $p = mv$,

which is the familiar equation $F = ma$ -- but it is worth being clear that $F = ma$ is only a special case of the more general momentum form of the second law, valid specifically when the mass of the body does not change with time. In situations where mass itself varies (a rocket burning and ejecting fuel, for instance, is exactly such a case, taken up again in Section 4.6), the momentum form $F = dp/dt$ must be used directly, because $F = ma$ on its own would miss the contribution to force coming from the changing mass. …