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Transfering an Electrical Concept



The maximum power theorem is well known to electrical engineers.
It may be expressed like this:
Maximal power is transfered when the source impedance equals
the characteristic impedance of the output.

This is easily illustrated in concrete examples, such as the
12 volt car battery driving the 6 volt starter motor to spin the
auto engine fastest.

The other day I mentioned on a newsgroup, the lessons that
could be drawn from a TV depiction of a junkyard competition,
in this case to maximize the range achieved by a football
impelled by some impromptu device.

The two devices shown in contention were a pneumatic cannon driven
by a pedal powered air compressor, a quick release valve,
and a length of 8 inch (?) pipe on the one side.

On the other side, a crossbow composed by two elliptical
leaf springs attached to a rear axle, with draw strings connected
from both leaves to a ball cage.

The air gun blew the ball 30 meters, but the crossbow could make
only a ten meter range.

I remarked that the mechanical impedences were not well matched
in the cross bow between springs and ball mass. The concept is
evidently much less known in mechanical terms than in electrical
terms. When asked, I found it difficult to provide a convincing
explanation of matching mechanical impedences.

Maximizing power to a ball evidently does not imply maximizing its
kinetic energy. (For in that case, a very massive projectile
dribbling out of the crossbow would have transferred most of the
available stored energy to the projectile.)

Looking through mechanics texts, I find a proof that the jet speed
onto a Pelton wheel transfers most power when the bucket speeds
are half the jet speed, but that doesn't mention the impedance
matching concept.

What is a reasonable explanation for the need to choose an optimal
bolt mass in a crossbow for best range?
Impulse? Change of momentum?

Thanks in advance.

Brian

brian whatcott <inet@intellisys.net> Altus OK
Eureka!