Page 47 - Mechanical Engineers' Handbook (Volume 2)

36 Input and Output Characteristics

3. The striker is the more massive of the two, m 1, 1. Then the striker, M , will
1
follow at reduced velocity behind the struck ball after their impact, and the struck
ball will move away faster than the initial velocity of the striker (because it has less
mass). Again, the initial energy is shared between the balls.
Thus, the initial energy is conserved in all of these transactions. But the energy can be
transferred completely from one ball to the other if and only if the two balls have the same
mass.
If these balls were made of clay so that the impact was perfectly plastic (no rebound
whatsoever), then 0, so the striker and struck balls would move off together at the same
velocity after impact no matter what the masses of the two balls. They would be effectively
stuck together. The final momentum of the pair would equal the initial momentum of the
striker because, on a frictionless surface, there are no external forces acting, but energy could
not be conserved because of the losses in plastic deformation during the impact. The final
velocities for the same three cases are
1
v v (5)
ƒ
1 m i
Since the task at hand, however, is to transfer kinetic (KE) from the first ball to the
second, we are interested in maximizing the energy in the second ball after impact with
respect to the energy in the first ball before impact:

1
KE (M ,after) –M (v ) 2 M (1/(1 m)) (v ) 2 m
2
2
2
2ƒ
2
1i
2
1
KE (M ,before) –M (v ) 2 M (v ) 2 (1 m) 2 (6)
1i
2
1
1
1i
1
This takes on a maximum value of – 1 4 when m 1 and falls off rapidly as m departs
from 1.
Thus, after the impact of two clay balls of equal mass, one-fourth of the initial energy
remains in the striker, one-fourth is transferred to the struck ball, and one-half of the initial
energy of the striker is lost in the impact. If the struck ball is either larger or smaller than
the striker, however, then a greater fraction of the initial energy is dissipated in the impact
and a smaller fraction is transferred to the second ball. The reader should reflect on how
this influences the severity of automobile accidents between vehicles of different sizes.
2.3 A Human Example
Those in good health can try the following experiment. Run up a long flight of stairs one at
a time and record the elapsed time. After a rest, try again, but run the stairs two at a time.
Still later, try a third time, but run three steps at a time. Most runners will find that their
best time is recorded for two steps at a time.
In the first test, the runner’s legs are velocity limited: Too much work is expended
simply moving legs and feet, and the forces required are too low to use the full power of
the legs effectively. In the third test, although the runner’s legs do not have to move very
quickly, they are on the upper edge of their force capabilities for continued three-step jumps;
the forces required are too high and the runner could, at lower forces, move his or her legs
much faster. In the intermediate case there is a match between the task and the force–velocity
characteristics of the runner’s legs.
Bicycle riders assure this match with a variable-speed transmission that they adjust so
they can crank the pedals at approximately 60 RPM. We will later look at other means of
ensuring the match between source capabilities and load requirements when neither of them