Heat seeking missiles

[BMAD]

Four heat-seeking missiles are initially placed at the corners of a square with side length s. Each missile flies at a constant speed toward the missile on its left. Describe the path each missile takes until it collides with the rest in the square's center. What is this path's length?

What about five missiles placed on a regular pentagon? n missiles on a regular n-gon?

[bonanova]

Standard puzzle: Four ants on the corners of a unit square walk toward the ant on their right. What is their path length? Path length is 1, same as if the target ant is stationary. This is because as the ants move as an ensemble they remain as four corners of a (shrinking) square. Each target ant moves, but neither toward nor away from his pursuer.

The heat-sinking missiles act like the ants and thus travel a distance s.

The shape of their path is a spiral of some type.

For n-gons, (n different from 4) the interior angles a differ from 90^o. The target thus moves away from (n>4) or toward (n<4) the pursuer by a fraction f = sin (a-90^o) of the distance its pursuer moves toward it. The missiles collide in any case, but after traveling a distance s multiplied by a factor (1+f). Obviously, the path length increases with n. It approaches infinity as the n-gon becomes similar to a circle, and it decreases to s/2 for the degenerate 2-gon (line segment.)

The whole story:

s multiplier n sides

---------------------

0.50 2

0.67 3

1.00 4

1.45 5

2.00 6

2.66 7

3.41 8

5.24 10

7.46 12

20.4 20

127 50

507 100

2026 200

12665 500

50661 1000

202645 2000

1266515 5000

5066201 10000