Applications of Reduction
The “Many
Hands” Theorem:
Assume that r1t1
= r2t2 = d. Then:
d/(r1+r2) = (t1<+>t2)
d/(t1+t2) = (r1<+>r2)
d/(r1-r2) = (t1<->t2)
d/(t1-t2) = (r1<->r2)
In a rate-time
problem, when rates add, times reduce; and when rates subtract, times
protract. “Many hands make light work.”
If Alice can
trim the hedges in A hours of work, and Bob can trim the hedges in B hours,
then if they work together, and if their work rates add, then they will finish
the job in A<+>B hours.
Two ships
cross the same ocean, starting at the same time and heading for each other's
home port. Let F be the time that the first ship took to go from port A to port
B, let S be the time that the second ship took to go from port B to port A, and
let M be the time it took for them to meet in mid-ocean. Approach-rates add, so
times reduce: M equals F <+> S.
Two racecars,
when driving towards each other from a distance of a mile, pass each other in P
seconds; when the faster car pursues the slower car from one mile behind, it
takes C seconds to catch up. How quickly does each car run a mile?
Answer:
Respectively, 2(P<+>C) and
2(P<->C) seconds.
If you drive a
mile at velocity v1, and then another mile at velocity v2,
then on average you will go at velocity 2(v1 <+> v2);
the harmonic mean.
Kirkhoff's Law
for parallel resistors goes:
R12P = R1
<+> R2 .
Resistances
reduce in parallel.
Two simple
lenses with focal lengths f1 and f2, when placed
together, form a compound lens with focal length f1 <+> f2
.
A lens made of
material with index of refraction n, and with lens radii r1 and r2,
has focal length (r1 <->
r2)/(n-1) .
D
.
. . |
. . |
. . |
. . |
h
. . |
. . |
. a . b |
A x
B C
In this
diagram, h = x
(tan(a) <-> tan(b))
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