You are probably familiar with a puzzle which is usually tricked out
with plausible numbers which conjure an image of a bird/fly/insect
flying repeatedly between two trains approaching each other
on the same line.
This may be expressed as
train1 speed x mph train2 speed y mph bird speed z mph
Trains' starting separation a miles.
Question: How far does the bird travel between the two trains
before they collide?
This can be a vivid illustration of the benefit of a little analysis
in avoiding extended calculations, which goes like this:
joint speed x + y mph; time to collide a/(x+y) hours;
Distance flown by bird; z.a/(x+y) miles
There is a comparable puzzle called four birds in a square field.
If each bird sets off at the same time, and aims for the next bird in
clockwise progression, how far does each bird fly before the birds collide?
For a square of side a miles, and an airspeed of x mph,
the mutual curvilinear approach speed this time is just x mph - because
always flies orthogonally.
Distance flown by a bird is a miles.
There is a third variation, which goes like this:
There is a bird at the corner of an equilateral triangular field of side
If each bird sets off flying at x mph at the same time, and aims
directly for the next bird in clockwise progression, how far does each
bird fly before the joint collision?
The method should not take a giant leap of intuition, if it is prepared
by these prior examples....