Momentum, Force, and Newton's Second Law

In a closed system (one that does not exchange any matter with the outside and is not acted on by outside forces), the total momentum is constant . This fact, known as the law of conservation of momentum, is implied by Newton's laws of motion. Suppose, for example, that two particles interact. Because of the third law, the forces between them are equal and opposite. If the particles are numbered 1 and 2, the second law states that

$\frac{d p_1}{d t} = - \frac{d p_2}{d t}$

or

$\frac{d}{d t} \left(p_1+ p_2\right)= 0$

Therefore, total momentum (p₁+p₂) is constant. If the velocities of the particles are u₁ and u₂ before the interaction, and afterwards they are v₁ and v₂, then

$m_1 u_{1} + m_2 u_{2} = m_1 v_{1} + m_2 v_{2}$

This law holds regardless of the nature of the interparticle (or internal) force, no matter how complicated the force is between particles. Similarly, if there are several particles, the momentum exchanged between each pair of particles adds up to zero, so the total change in momentum is zero.

Newton's Second Law

Newton actually stated his second law of motion in terms of momentum: The net external force equals the change in momentum of a system divided by the time over which it changes. Using symbols, this law is

$F_{net} = \frac{\Delta p}{\Delta t}$ ,

where $F_{net}$ is the net external force, Δp is the change in momentum, and Δt is the change in time.

This statement of Newton's second law of motion includes the more familiar $F_{net} = ma$ as a special case. We can derive this form as follows. First, note that the change in momentum Δp is given by Δp=Δ(mv). If the mass of the system is constant, then Δ(mv)=mΔv. So for constant mass, Newton's second law of motion becomes

$F_{net} = \frac{\Delta p}{\Delta t} = \frac{m\Delta v}{\Delta t}$ .

Because Δv/Δt=a, we get the familiar equation $F_{net} = ma$ when the mass of the system is constant. Newton's second law of motion stated in terms of momentum is more generally applicable because it can be applied to systems where the mass is changing, such as rockets, as well as to systems of constant mass.

Momentum in a Closed System

In a game of pool, the system of entire balls can be considered a closed system. Therefore, the total momentum of the balls is conserved.