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306 CHAPTER 10 Systems of Particles

? What is equation of motion of a system of particles, and to what extent does
the translational motion of a jumper resemble projectile motion? (Page 324 in
Section 10.3)
o far we have dealt almost exclusively with the motion of a single particle. Now we
Swill begin to study systems of particles interacting with each other via some forces.
This means we must examine, and solve, the equations of motion of all these particles
simultaneously.
Since chunks of ordinary matter are made of particles—electrons, protons, and
neutrons—all the macroscopic bodies that we encounter in our everyday environment
are in fact many-particle systems containing a very large number of particles. However,
for most practical purposes, it is not desirable to adopt such an extreme microscopic
point of view, and in the preceding chapters we treated the motion of a macroscopic
body, such as an automobile, as motion of a particle. Likewise, in dealing with a system
consisting of several macroscopic bodies, we will often find it convenient to treat each of
these bodies as a particle and ignore the internal structure of the bodies. For example,
when investigating a collision between two automobiles, we may find it convenient to
pretend that each of the automobiles is a particle—we then regard the colliding auto-
mobiles as a system of two particles which exert forces on each other when in contact.
And when investigating the Solar System, we may find it convenient to pretend that
each planet and each satellite is a particle—we then regard the Solar System as a system
of such planet and satellite particles loosely held together by gravitation and orbiting
around the Sun and around each other.
The equations of motion of a system of several particles are often hard, and some-
times impossible, to solve. It is therefore necessary to make the most of any informa-
tion that can be extracted from the general conservation laws. In the following sections
we will become familiar with the momentum vector, and we will see how the laws of con-
servation of momentum and of energy apply to a system of particles.

10.1 MOMENTUM

Newton’s laws can be expressed very neatly in terms of momentum, a vector quantity
of great importance in physics. The momentum of a single particle is defined as the prod-
1
uct of the mass and the velocity of the particle:

momentum of a particle p mv (10.1)

Thus, the momentum p is a vector that has the same direction as the velocity vector,
but a magnitude that is m times the magnitude of the velocity.The SI unit of momen-
tum is kg m/s; this is the momentum of a mass of 1 kg when moving at 1 m/s.
The mathematical definition of momentum is consistent with our intuitive, every-
day notion of “momentum.” If two cars have equal masses but one has twice the veloc-
ity of the other, it has twice the momentum. And if a truck has three times the mass
of a car and the same velocity, it has three times the momentum. During the nine-
teenth century physicists argued whether momentum or kinetic energy was the best
measure of the “amount of motion” in a body. They finally decided that the answer

1
The momentum p mv is sometimes referred to as linear momentum to distinguish it from angular momen-
tum, discussed in Chapter 13.