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324 CHAPTER 10 Systems of Particles

m v m v m v
1 x,1
n x,n
2 x,2
v x,CM
M
Note that this equation has the same mathematical form as Eq. (10.18); that is, the
velocity of the center of mass is an average over the particle velocities, and the number
of times each particle velocity is included is directly proportional to its mass.
Since similar equations apply to the y and z components of the velocity, we can
write a vector equation for the velocity of the center of mass:

m v m v m v
velocity of the center of mass v 1 1 2 2 n n (10.37)
CM
M
The quantity in the numerator is simply the total momentum [compare Eq. (10.1)];
hence Eq. (10.37) says

P
v CM (10.38)
M
or

momentum in terms of velocity of CM P M v (10.39)
CM
This equation expresses the total momentum of a system of particles as the product of
the total mass and the velocity of the center of mass. Obviously, this equation is anal-
ogous to the familiar equation p mv for the momentum of a single particle.
We know, from Eq. (10.13), that the rate of change of the total momentum equals
the net external force on the system,

d P
F
dt ext

If we substitute P Mv and take into account that the mass is constant, we find
CM
d P d dv CM
(M v ) M Ma
dt dt CM dt CM
and consequently
motion of center of mass Ma CM F ext (10.40)

This equation for a system of particles is the analog of Newton’s equation for
motion for a single particle. The equation asserts that the center of mass moves as
though it were a particle of mass M under the influence of a force F .
ext
Concepts This result justifies some of the approximations we made in previous chapters. For
in instance, in Example 9 of Chapter 2 we treated a diver falling from a cliff as a parti-
Context
cle. Equation (10.40) shows that this treatment is legitimate: the center of mass of the
diver, under the influence of the external force (gravity), moves with a downward accel-
eration g, just as though it were a freely falling particle. Likewise, after a high jumper
leaves the ground, his center of mass moves along a parabolic trajectory, as though it
were a projectile, and the shape and height of this parabolic trajectory is unaffected
by any contortions the high jumper might perform while in flight. From Chapter 4,
we know that the initial vertical velocity v determines the maximum height h of the
y
center of mass; that is, v 12gh .The contortions of the jumper enable his body to
y
pass over a bar roughly 10 cm above the maximum height of the center of mass.