Two particles each of mass m are connected by a light inextensible string and a particle of mass M is attached to the midpoint of the string. The system is at rest on a smooth horizontal table with string just taut and in a straight line. The particle M is given a velocity v along the table perpendicular to the string. Prove that when the two particles are about to collide :
(i) The velocity of M is `(Mv)/((M + 2m))`
(ii) The speed of each of the other particles is `((sqrt2M (M + m))/((M + 2m)))v`.

Explanation

To solve the problem step by step, we will analyze the system of particles and apply the principles of conservation of momentum and energy.

### Step 1: Understand the System
We have two particles, each of mass \( m \), connected by a light inextensible string. A particle of mass \( M \) is attached to the midpoint of the string. Initially, the system is at rest, and then the particle \( M \) is given a velocity \( v \) perpendicular to the string.

### Step 2: Apply Conservation of Linear Momentum
When particle \( M \) moves with velocity \( v \), it will exert forces on the two particles of mass \( m \). Since the system is isolated, the total momentum before and after must be conserved.

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