IIT-JEE 2009 Multiple Choice Question (Single Answer Type) on Elastic Collision

The following question which appeared in IIT-JEE 2009 question paper is based on the principle that in an elastic collision between two particles of the same mass, the velocities get interchanged: 

Two small particles of equal masses start moving in opposite directions from a point A in a horizontal circular orbit. Their tangential velocities are v and 2v respectively as shown in the figure. Between collisions, the particles move with constant speeds. After making how many elastic collisions, other than that at A, these two particles will again reach the point A?

(a) 4

(b) 3

(c) 2

(d) 1    

Particle P₁, after traversing one third (AB) of the circular track in the anticlockwise direction, will collide with particle P₂ at position B. This collision occurs when the particle P₂  has traversed two thirds (ACB) of the circular path (since the speed of P₂ is twice that of P₁). 

         Since the collision is elastic and the particles are of equal masses, their velocities are interchanged. P₁ now travels in the clockwise direction with speed 2v and P₂ travels in the anticlockwise direction with speed v. The second collision takes place at position C. (BAC is two thirds of the circular path while BC is one third of the path). The second collision at C results in the reversal of the velocities and P₁ now travels in the anticlockwise direction with speed v where as P₂ travels in the clockwise direction with speed 2v. Therefore, the third collision will occur at A.

Since the third collision at A is not to be counted, the answer is 2 [Option (c)].

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Now suppose we modify the above question as follows:

Two small particles of equal masses start moving in opposite directions from a point A in a horizontal circular frictionless track. Their speeds are v and 3v respectively as shown in the figure. After making how many elastic collisions, other than that at A, these two particles will again reach the point A ?

(a) 4

(b) 3

(c) 2

(d) 1

[You may use the figure shown with the question above, replacing the velocity 2v with 3v]

You can easily arrive at the answer which is 3 [Option (b)].