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10.1 Momentum 309

This means that the sum p p is a constant of the motion:
1 2

p p [constant] (10.5) momentum conservation for two particles
1 2
This is the Law of Conservation of Momentum. Note that Newton’s Third Law is an
essential ingredient for establishing the conservation of momentum: the total momen-
tum is constant because the equality of action and reaction keeps the momentum changes
of the two particles exactly equal in magnitude but opposite in direction—the particles
merely exchange some momentum by means of their mutual forces. Thus, for our par-
ticles, the total momentum P at some instant equals the total momentum P at some
other instant, so
P P

Conservation of momentum is a powerful tool which permits us to calculate some
general features of the motion even when we are ignorant of the detailed properties of
the interparticle forces.The following examples illustrate how we can use conservation
of momentum to solve some problems of motion.

(a)
A gun used onboard an eighteenth-century warship is mounted
EXAMPLE 2 m 1 m
on a carriage which allows the gun to roll back each time it is 2
fired (Fig. 10.3).The mass of the gun, including the carriage, is 2200 kg.The gun
fires a 6.0-kg shot horizontally with a velocity of 500 m/s. What is the recoil veloc-
ity of the gun? x

SOLUTION: The total momentum of the shot plus the gun must be the same before
the firing and just after the firing. Before, the total momentum is zero (Fig. 10.3a): (b)

P 0 m
v' 2 2 m 1 v' 1
After, the (horizontal) velocity of the shot is v , and the velocity of the gun is v (as
1 2
above, the primes on mathematical quantities indicate that these are evaluated after
the firing); hence the total momentum is x

P m v m v Gun recoils Shot is fired
1 1 2 2
horizontally. horizontally.
where m 6.0 kg is the mass of the shot and m 2200 kg is the mass of the gun
1 2
(including the carriage). Thus, momentum conservation tells us FIGURE 10.3 (a) Initially, the gun and
0 m v m v the shot are at rest. (b) After the firing, the
1 1 2 2 gun recoils toward the left (the velocity v of
2
or the gun is negative).
m 1
v v
2 m 1
2
The negative sign indicates that v , the recoil velocity of the gun, is opposite to
2
the velocity of the shot and has a magnitude
m 1
v v
2 m 1
2
6.0 kg
500 m/s 1.4 m/s
2200 kg

COMMENTS: Note that the final velocities are in the inverse ratio of the masses:
the shot emerges with a large velocity, and the gun rolls back with a low velocity.