center of mass frame. Chapter 7 discusses Einstein’s resolution
of this problem, but the basic point is that youcan’tride the mo-
torcycle alongside the light beam, because material objects can’t
go as fast as the speed of light. A beam of light has no center of
mass frame of reference.
Discussion Questions
A Make up a numerical example of two unequal masses moving in one
dimension at constant velocity, and verify the equationpt ot al=mt ot alvcm
over a time interval of one second.
B A more massive tennis racquet or baseball bat makes the ball fly
off faster. Explain why this is true, using the center of mass frame. For
simplicity, assume that the racquet or bat is simply sitting still before the
collision, and that the hitter’s hands do not make any force large enough
to have a significant effect over the short duration of the impact.3.1.7 Totally inelastic collisions
On p. 139 we discussed collisions that were totally elastic (no
conversion of KE into other types of energy). A useful application
of the center of mass frame of reference is to the description of the
opposite extreme, a totallyinelasticcollision.
A totally inelastic collision cannot just be defined as one in which
all the KE is converted into other forms, both because the defini-
tion would depend on our frame of reference and because there is
a constraint imposed by conservation of momentum. Let’s say that
a golfer hits a ball. In the frame of reference of the grass, it would
violate conservation of momentum if the ball were to stay put while
the club simply stopped moving. If such a complete cessation of
motion is to happen, then it must occur in the center of mass frame
of reference. In the c.m. frame, there is zero total momentum both
before and after the collision. Thus if we observe no motion at all
after the collision, we must be in the c.m. frame.
Therefore we define a totally inelastic collision as one in which
there is no motion in the c.m. frame in the final state. An observer
watching such a collision, in any frame, will see that the amount of
KE transformed into other forms of energy is as great as possible
subject to conservation of momentum.
When objects touch physically (possibly crumpling or changing
shape during the collision) in a totally elastic collision, the final state
in the c.m. frame is one in which the two objects are at rest and
touching. In other frames of reference, we see the objects stick to
each other and travel away together after the collision. An example
of this type was example 11 on p. 138, in which one car rear-ended
another, and they stuck together as a unit after the crash.148 Chapter 3 Conservation of Momentum