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10.4 Energy of a System of Particles 327

COMMENT: Note that to relate both points of impact to the center of mass, we had
to know that the impacts occurred at the same instant; we must always use the
coordinates of a system of particles at a particular instant when calculating the
center of mass.

✔ Checkup 10.3

QUESTION 1: When you crawl from the rear end of a canoe to the front end, the boat
moves backward relative to the water. Explain.
QUESTION 2: You are locked inside a boxcar placed on frictionless wheels on railroad
tracks. If you walk from the rear end of the boxcar to the front end, the boxcar rolls back-
ward. Is it possible for you to make the boxcar roll a distance longer than its length?
QUESTION 3: You drop a handful of marbles on a smooth floor, and they bang into
each other and roll away in all directions. What can you say about the motion of the
center of mass of the marbles after the impact on the floor?
QUESTION 4: An automobile is traveling north at 25 m/s. A truck with twice the mass
of the automobile is heading south at 20 m/s. What is the velocity of the center of
mass of the two vehicles?
(A) 0 (B) 5 m/s south (C) 5 m/s north
(D) 10 m/s south (E) 10 m/s north

10.4 ENERGY OF A SYSTEM OF PARTICLES

The total kinetic energy of a system of particles is simply the sum of the individual
kinetic energies of all the particles,

1 2 1 2 1 2
K m v m v m v (10.42) kinetic energy of a system of particles
2 1 1 2 2 2 2 n n
Since Eq. (10.39) for the momentum of a system of particles resembles the expres-
sion for the momentum of a single particle, we might be tempted to guess that the
equation for the kinetic energy for a system of particles also can be expressed in the form
1 2
of the translational kinetic energy of the center of mass Mv CM ,resembling the kinetic
2
energy of a single particle. But this is wrong! The total kinetic energy of a system of par-
1 2
ticles is usually larger than Mv CM . We can see this in the following simple example:
2
Consider two automobiles of equal masses moving toward each other at equal speeds.The
1 2
velocity of the center of mass is then zero,and consequently Mv 0.However,since
2 CM
each automobile has a positive kinetic energy, the total kinetic energy is not zero.
If the internal and external forces acting on a system of particles are conservative,
then the system will have a potential energy. We saw above that for the specific exam-
ple of the gravitational potential energy near the Earth’s surface, the potential energy
of the system took the same form as for a single particle, U Mgy [see Eq. (10.33)].
CM
But this form is a result of the particular force (uniform and proportional to mass); in
general, the potential energy for a system does not have the same form as for a single
particle. Unless we specify all of the forces, we cannot write down an explicit formula
for the potential energy; but in any case, this potential energy will be some function of
the positions of all the particles. The total mechanical energy is the sum of the total