11.2.1 Systems of Particles

Equation 11.18 illustrates the idea of a conservation law for momentum, but it was derived for the specific case of two objects colliding in one dimension. Our goal is to develop a more general law of conservation of momentum, a law that will be valid in three dimensions and that will work for any type of interaction. The next few paragraphs are fairly mathematical, so you might want to begin by looking ahead to Equations 11.26 and the statement of the law of conservation of momentum to see where we’re heading.

Our study of energy in the last two chapters has emphasized the importance of having a clearly defined system. The same is true for momentum. Consider a system consisting of N particles. FIGURE 11.11 shows a simple case where $N = 3$ , but N could be anything. The particles might be large entities (cars, baseballs, etc.), or they might be the microscopic atoms in a gas. We can identify each particle by an identification number k. Every particle in the system interacts with every other particle via action/reaction pairs of forces $\vec{F}_{j on k}$ and $\vec{F}_{k on j}$ . In addition, every particle is subjected to possible external forces $\vec{F}_{ext on k}$ from agents outside the system.

The figure illustrates a system of three particles interacting with each other. The figure also shows the forces associated with the interactions.

FIGURE 11.11 Full Alternative Text

If particle k has velocity $\vec{v}_{k}$ , its momentum is $\vec{p}_{k} = m_{k} \vec{v}_{k}$ . We define the total momentum $\vec{P}$ of the system as the vector sum

\vec{P} = total momentum = {\vec{p}}_{1} + {\vec{p}}_{2} + {\vec{p}}_{3} + \dots + {\vec{p}}_{N} = \sum_{k = 1}^{N} {\vec{p}}_{k}

(11.19)

That is, the total momentum of the system is the vector sum of the individual momenta.

The time derivative of $\vec{P}$ tells us how the total momentum of the system changes with time:

\frac{d \vec{P}}{d t} = \sum_{k}^{} \frac{d \vec{p}_{k}}{d t} = \sum_{k}^{} \vec{F}_{k}

(11.20)

where we used Newton’s second law for each particle in the form $\vec{F}_{k} = d \vec{p}_{k} / d t$ , which was Equation 11.4.

The net force acting on particle k can be divided into external forces, from outside the system, and interaction forces due to all the other particles in the system:

\vec{F}_{k} = \sum_{j \neq k}^{} \vec{F}_{j on k} + \vec{F}_{ext on k}

(11.21)

The restriction $j \neq k$ expresses the fact that particle k does not exert a force on itself. Using this in Equation 11.20 gives the rate of change of the total momentum $\vec{P}$ of the system:

\frac{d \vec{P}}{d t} = \sum_{k}^{} \sum_{j \neq k}^{} \vec{P}_{j on k} + \sum_{k}^{} \vec{F}_{ext on k}

(11.22)

The double sum on $\vec{F}_{j on k}$ adds every interaction force within the system. But the interaction forces come in action/reaction pairs, with $\vec{F}_{k on j} = - \vec{F}_{j on k}$ , so ${\vec{F}}_{k on j} + {\vec{F}}_{j on k} = \vec{0}$ . Consequently, the sum of all the interaction forces is zero. As a result, Equation 11.22 becomes

\frac{d \vec{P}}{d t} = \sum_{k}^{} \vec{F}_{ext on k} = \vec{F}_{net}

(11.23)

where $\vec{F}_{net}$ is the net force exerted on the system by agents outside the system. But this is just Newton’s second law written for the system as a whole! That is, the rate of change of the total momentum of the system is equal to the net force applied to the system.

A photograph of a space rocket moving upward while its boosters expel burning gases from the bottom exhaust. — The total momentum of the rocket $+$ gases system is conserved, so the rocket accelerates forward as the gases are expelled backward.

Equation 11.23 has two very important implications. First, we can analyze the motion of the system as a whole without needing to consider interaction forces between the particles that make up the system. In fact, we have been using this idea all along as an assumption of the particle model. When we treat cars and rocks and baseballs as particles, we assume that the internal forces between the atoms—the forces that hold the object together—do not affect the motion of the object as a whole. Now we have justified that assumption.