University Physics I - Classical Mechanics, 2019

56 CHAPTER 3. MOMENTUM AND INERTIA
of the standard. Suppose that we have two objects, to which we have assigned masses m 1 and m 2
by arranging for each to collide with the “standard object” independently. If we now arrange for
a collision between objects 1 and 2 directly, will we actually find that the ratio of their velocity
changes is given by the ratio of the separately determined masses m 1 and m 2 ? We certainly would
need that to be the case, in order for the concept of inertia to be truly useful; but again, we should
not assume anything until we have tested it! Fortunately, the tests would indeed reveal that, in
every case, the expected relationship holds 3 − Δv 2
= m 1
(3.2)
Δv 1 m 2
At this point, we are not just in possession of a useful definition of inertia, but also of a veritable
law of nature, as I will explain next.
3.2 Momentum
For an object of (inertial) mass m moving, in one dimension, with velocity v, we define its momentum
as
p = mv (3.3)
(the choice of the letter p for momentum is apparently related to the Latin word “impetus”).
We can think of momentum as a sort of extension of the concept of inertia, from an object at rest
to an object in motion. When we speak of an object’s inertia, we typically think about what it
may take to get it moving; when we speak of its momentum, we typically think of that it may take
to stop it (or perhaps deflect it). So, both the inertial mass m and the velocity v are involved in
the definition.
We may also observe that what looks like inertia in some reference frame may look like momentum
in another. For instance, if you are driving in a car towing a trailer behind you, the trailer has
only a large amount of inertia, but no momentum, relative to you, because its velocity relative to
you is zero; however, the trailer definitely has a large amount of momentum (by virtue of both its
inertial mass and its velocity) relative to somebody standing by the side of the road.
3.2.1 Conservation of momentum; isolated systems
For a system of objects, we treat the momentum as an additive quantity. So, if two colliding objects,
of masses m 1 and m 2 , have initial velocities v 1i and v 2i , we say that the total initial momentum of
3 Equation 3.2 actually is found to hold also at the microscopic (or quantum) level, although there we prefer to
state the result by saying that conservation of momentum holds (see the following section).