Conservation
of Momentum
What happens when a group
of particles are all interacting? Qualitatively speaking, each exerts equal and
opposite impulses on the other, and though the individual momentum of any given
particle might change, the total momentum of the system remains constant. This
phenomenon of momentum constancy describes the conservation of linear momentum
in a nutshell; in this section we shall prove the existence of the conservation
of energy by using what we already know about momentum and systems of
particles.
Momentum
in a System of Particles
Just as we first defined
kinetic energy for a single particle, and then examined the energy of a system,
so shall we now turn to the linear momentum of a system of particles. Suppose
we have a system of N particles, with masses m 1, m 2,…, m n .
Assuming no mass enters or leaves the system, we define the total momentum of
the system as the vector sum of the individual momentum of the particles:
|
P
|
=
|
p 1 + p 2 + ... + p n
|
|
|
|
=
|
m 1 v 1 + m 2 v 2 + ... + m n v n
|
|
Recall from our discussion of center of mass that:
v cm =
(m 1 v 1 + m 2 v 2 + ... + m n v n)
where M is
the total mass of the system. Comparing these two equations we see that:
|
P = Mv cm
|
|
Thus
the total momentum of the system is simply the total mass times the velocity of
the center of mass. We can also take a time derivative of the total momentum of
the system:
Recall
also that, for a system of particles,
Clearly,
then:
|
|
|
Don't
worry if the calculus here is complex. Though our definition of the momentum of
a system of particles is important, the derivation of this equation only
matters because it tells us a great deal about momentum. When we explore this
equation further we will generate our principle of conservation of linear
momentum.
Conservation
of Linear Momentum
From our last equation we
will consider now the special case in which
F ext = 0 . That is, no
external forces act upon an isolated system of particles. Such a situation
implies that the rate of change of the total momentum of a system does not
change, meaning this quantity is constant, and proving the principle of the
conservation of linear momentum:
When
there is no net external force acting on a system of particles the total
momentum of the system is conserved.
It's that simple. No matter
the nature of the interactions that go on within a given system, its total
momentum will remain the same. To see exactly how this concept works we shall
consider an example.
Conservation
of Linear Momentum in Action
Let's consider a cannon
firing a cannonball. Initially, both the cannon and the ball are at rest.
Because the cannon, the ball, and the explosive are all within the same system
of particles, we can thus state that the total momentum of the system is zero.
What happens when the cannon is fired? Clearly the cannonball shoots out with
considerable velocity, and thus momentum. Because there are no net external
forces acting on the system, this momentum must be compensated for by a
momentum in the opposite direction as the velocity of the ball. Thus the cannon
itself is given a velocity backwards, and total momentum is conserved. This
conceptual example accounts for the "kick" associated with firearms.
Any time a gun, a cannon, or an artillery piece releases a projectile, it must
itself move in the direction opposite the projectile. The heavier the firearm,
the slower it moves. This is a simple example of the conservation of linear
momentum.
By both examining the
center of mass of a system of particles, and developing the conservation of
linear momentum we can account for a great deal of motion in a system of
particles. We now know how to calculate both the motion of the system as a
whole, based on external forces applied to the system, and the activity of the
particles within the system, based on momentum conservation within the system.
This topic, dealing with momentum, is as important as the last one, dealing
with energy.
Both concepts are universally applied: while Newton's
Laws apply
only to mechanics, conservation of momentum and energy are used in relativistic
and quantum calculations as well.
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