# "n" no. of particles are located at the vertices of a regular polygon of "n" sides having the edge length "a". They all start moving simultaneously with equal constant speed "v" heading towards the particle next to it all the time. How long will the particles take to collide?

Aman Bansal
592 Points
12 years ago

Dear Pooja,

Thus, while in motion, the turtles form regular N-gons that share the same center. With time, the polygons rotate and shrink. Lets place an observer at the center of the polygons and make him rotate along with the turtles at their angular speed. What will such an observer see? He will see the turtles approaching him on straight lines, albeit moving sideways. The motion of the turtles along the radial lines is uniform. Indeed, their velocities that are directed along the side lines of the polygon have fixed projections on the radial lines joining the center with the vertices.

Let OA and OB be two such lines. AB is the side length at time t = 0, S = AB; OB is the distance to the center, Rr = OB; H is the midpoint of AB. From DBOH,

S/2 = R·sin(a),

where a = p/N. The projection of velocity V on OB can be found from DBCD, where CD is perpendicular to OB. Taking V = |V| = BC,

BD = V·cos(p/2 - a) = V·sin(a).

Thus from the view point of the rotating observer, the turtles move along straight lines, have to travel distance R at the speed of V·sin(a). The time need to accomplish this task is

(1)
 T = R/[V·sin(a)] = S/[2V·sin2(a)] = S/[V(1 - cos(2a))].

Thus the turtles will meet in time T given by (1). Since they move with the constant speed V, in time T they will travel the distance of D = VT:

D = S/[1 - cos(2a)].

For N = 4, 2a = 2p/4 = p/2, so that cos(2a) = 0 and D = S, which explains the popularity of this case.

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Thanks

Aman Bansal