Page 46 - Mechanical Engineers' Handbook (Volume 2)

2 Familiar Examples of Input-Output Interactions 35

2.2 Energy Exchange
Interactions between systems are not restricted to resistive behavior, nor is the concept of
impedance matching restricted to real, as opposed to reactive, impedances. Consider a pair
of billiard balls on a frictionless table (to avoid the complexities of spin), and consider that
their impact is governed by a coefficient of restitution, . Before impact, only one ball is
moving, but afterward both may be. The initial and final energies are as follows:
Initial energy –M v 2
1
2
1 1i
Final energy 1 –M v 2 –M v 2 (1)
1
2 2ƒ
1 1ƒ
2
2
where the subscript 1 refers to the striker and 2 to the struck ball, M is mass, v is velocity,
and the subscripts i and ƒ refer to initial and final conditions, respectively.
Since no external forces act on this system of two balls during their interaction, the
total momentum of the system is conserved:
M v M v M v M v
11i 22i 11ƒ 2 2ƒ
or
v mv v 1ƒ mv 2ƒ (2)
1i
2i
where m M /M . The second equation, required to solve for the final velocities, derives
2
1
from impulse and momentum considerations for the balls considered one at a time. Since
no external forces act on either ball during their interaction except those exerted by the other
ball, the impulses,* or integrals of the force acting over the time of interaction on the two,
are equal. (See impact in virtually any dynamics text.) From this it can be shown that the
initial and final velocities must be related:
(v v ) (v 1ƒ v ) (3)
2ƒ
2i
1i
where v 0 in this case and the coefficient of restitution is a number between 0 and 1.
2i
A 0 corresponds to a plastic impact while a 1 corresponds to a perfectly elastic impact.
Equations (2) and (3) can be solved for the final velocities of the two balls:
1 m 1
v 1ƒ v 1i and v 2ƒ v 1i (4)
1 m 1 m
Now assume that one ball strikes the other squarely† and that the coefficient of resti-
tution is unity (perfectly elastic impact). Consider three cases:
1. The two balls have equal mass, so m 1, and 1. Then the striking ball, M ,
1
will stop, and the struck ball, M , will move away from the impact with exactly the
2
initial velocity of the striking ball. All the initial energy is transferred.
2. The struck ball is more massive than the striking ball, m 1, 1. Then the striker
will rebound along its initial path, and the struck ball will move away with less than
the initial velocity of the striker. The initial energy is shared between the balls.

t Force dt, where Force is the vector sum of all the forces acting over the period of
*Impulse t 0
interaction, t.
†Referred to in dynamics as direct central impact.