102 CHAPTER 7. MOMENTUM
With this definition, then the left side of 7.10 is the initi al value of P and the right side is
the final value of P. So we have:
P
i
= P
f
(7.12)
that is, the value of the total momentum stay s the same, or as we say in physics, is conserved.
We need to recall the conditions under which we could make t his statement (our assump-
tions): We assumed that during the interaction, A and B were interacting with each other
but felt no forces from anything else. Another way to say this is that during the interaction,
the two objects were isolated; bec ause of that, their total momentum was conserved.
The principle can be made more general by includi ng more particles, and the lesson can
be stated as:
For a system of i solated particles, the total momentum is conserved.
7.1 .4 Collisi ons; Problems Using the Conservation of Momentum
Sometimes we are faced with a problem where there are two or interacting particles which
“feel” no forces from anything else, i.e. they are isolated. Sometimes we have a situation
where we are consideri ng the m otion of several particles over such a short period of time
that we can safely ignore the external forces but we can’t ignore the forces between the
particles. Then, for the purposes of the problem, the particles are (again) “isolated”. The
forces between t he particles may be very complicated, but that doesn’t matter. The total
momentum of the particles will stay the same before and after the interaction.
Examples of where such an interaction can occur are:
• When two particles travel freely, bounce off one another and then travel away from one
another in new directions; this is what we normally think of as a “collision”.
• When two particl es come together and stick to one another; the combined mass travels off
as one unit (with a mass equal to the sum of the individual masses).
• When a single mass explodes and the individual parts fly off in different dire ctions; we
would call this an “explosion”. Again, the sum of all the masses is the same be fore and after.
Though total momentum stays the same during these processes, what about the energy
of the particles? More specifically, since in the r apid interactions we are considering the
potential energy of the system doesn’t change by much we ask what happens to the kinetic
energy of the particles.
In general, the total kinetic energy can remain the same or decrease or even increase
for a collision; it depends on the nature of the interacting objects and the kind of force
the exe r t on each other. The usual case is that when real objec t s collide the force is i n
small part frictional in nature and then kinetic energy is lost, or rather it changes form to
thermal energy. But it is possible that the impact releases chemical or other stored energy
from an exploding element and then the kinetic energy would increase. But if the surfaces