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10.1 Momentum 307

depends on the context—as we will see in the examples in this chapter and the next,
sometimes momentum is the most relevant quantity, sometimes energy is, and some-
times both are relevant.
Newton’s First Law states that, in the absence of external forces, the velocity of a
particle remains constant. Expressed in terms of momentum, the First Law therefore
states that the momentum remains constant:

p [constant] (no external forces) (10.2) First Law in terms of momentum

Thus, we can say that the momentum of the particle is conserved. Of course, we could
equally well say that the velocity of this particle is conserved; but the deeper signifi-
cance of momentum will emerge when we study the motion of a system of several par-
ticles exerting forces on one another. We will find that the total momentum of such a
system is conserved—any momentum lost by one particle is compensated by a momen-
tum gain of some other particle or particles.
To express the Second Law in terms of momentum, we note that since the mass
is constant, the time derivative of Eq. (10.1) is

d p d v
m
dt dt
or
d p
ma
dt
But, according to Newton’s Second Law, ma equals the force; hence, the rate of change
of the momentum with respect to time equals the force:

d p
F (10.3) Second Law in terms of momentum
dt

This equation gives the Second Law a concise and elegant form.

A tennis player smashes a ball of mass 0.060 kg at a vertical
EXAMPLE 1
wall. The ball hits the wall perpendicularly with a speed of
40 m/s and bounces straight back with the same speed. What is the change of
momentum of the ball during the impact?
SOLUTION: Take the positive x axis along the direction of the initial motion of
the ball (see Fig. 10.1a). The momentum of the ball before impact is then in the
positive direction, and the x component of the momentum is

p mv 0.060 kg 40 m/s 2.4 kg m/s
x x
The momentum of the ball after impact has the same magnitude but the oppo-
site direction:

p 2.4 kg m/s
x
(Throughout this chapter, the primes on mathematical quantities indicate that
these quantities are evaluated after the collision.) The change of momentum is

p p p 2.4 kg m/s 2.4 kg m/s 4.8 kg m/s
x x x