Environmental Engineering Reference
In-Depth Information
momentum of the system is constant irrespective of the details of the internal forces
of the system. This very important result is known as the law of conservation of
momentum. For us, momentum conservation follows immediately as a consequence
of Newton's Second Law. However, that is to undermine the significance of what
is now understood to be a fundamental law of physics: momentum conservation
applies even in circumstances where Newton's Second Law does not
5
. Momentum
conservation will also play a very important role in Chapter 11. From the perspec-
tive of this chapter, it will provide a very powerful tool to help us solve problems
in dynamics. The beauty is that it can be used in circumstances where unknown
forces act within a system, for example when two objects collide elastically the
forces that are present during the impact are not generally known but since they are
internal to the system as a whole the total momentum of the system is unchanged.
It means that we can go ahead and compute the momentum of the system before
the action of the forces and then again afterwards, and the two must be the same.
Example 2.3.6
A cannon fires a cannonball at an angle θ to the horizontal and at
a speed v
0
relative to its muzzle. The cannon is constrained so that it recoils along
horizontal rails. Use momentum conservation to calculate the recoil velocity of the
cannon. Assume that the rails are frictionless.
Solution 2.3.6
Let the mass of the cannon be M, the mass of the cannonball m and
the recoil velocity be horizontal and of magnitude v. We will take the “system” to
be the cannon and cannonball. Since the rails are frictionless there are no exter-
nal forces acting in the horizontal direction and we can therefore use momentum
conservation in this direction. Naturally enough, we work in a frame of reference in
which the cannon and cannonball are both initially at rest. Momentum conservation
in the horizontal direction then gives
0
=
m(v
0
cos
θ
−
v)
−
Mv,
where the recoil speed of the cannon has been subtracted from the horizontal com-
ponent of the cannonball's velocity relative to the muzzle. This can be rearranged
to give
0
=
mv
0
cos
θ
−
v(M
+
m),
mv
0
cos
θ
m
i.e.
v
=
.
+
M
In this last example, the importance of the rails being frictionless should be
emphasised. Friction is to be viewed as an external force, for if it is not negligi-
ble then we would need to take account of the fact that some of the horizontal
momentum is transferred to the Earth. Momentum conservation would still apply
but now only if we widen the definition of our system to include the Earth. Just
how much recoil momentum is transferred, via friction, to the Earth requires an
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e.g. the study of quantum mechanical systems.
