A particle at rest suddenly disintegrates into two particles of equal masses which start moving. The two fragments will

Options

A. Move in the same direction with equal speeds
B. Move in any directions with any speed
C. Move in opposite directions with equal speeds
D. Move in opposite directions with unequal speeds

Answer

To solve the problem, we need to analyze the situation of a particle at rest that disintegrates into two particles of equal mass. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Initial Condition
The particle is initially at rest. This means its initial momentum is zero.

**Hint:** Remember that momentum is a vector quantity and is conserved in an isolated system.

### Step 2: Apply Conservation of Momentum
When the particle disintegrates into two equal masses, the total momentum of the system must remain zero (since it was initially at rest). If we denote the masses as \( m_1 \) and \( m_2 \) (both equal to \( m \)), and their velocities as \( v_1 \) and \( v_2 \), we can write the conservation of momentum as:

\[ m v_1 + m v_2 = 0 \]

This simplifies to:

\[ v_1 + v_2 = 0 \]

**Hint:** This equation indicates that the velocities of the two fragments must be equal in magnitude but opposite in direction.

### Step 3: Determine the Directions of Motion
Given that \( v_1 + v_2 = 0 \), if one particle moves in one direction (let's say to the right), the other must move in the opposite direction (to the left).

**Hint:** Think about how the center of mass behaves when the system is isolated and not influenced by external forces.

### Step 4: Equal Speeds of the Fragments
Since the masses are equal and the total momentum must remain zero, the two fragments must move with equal speeds. Therefore, if one fragment moves with speed \( v \), the other must also move with speed \( v \) but in the opposite direction.

**Hint:** The conservation of momentum ensures that the center of mass remains stationary if no external forces act on the system.

### Conclusion
The two fragments will move in opposite directions with equal speeds.

### Final Answer
The two fragments will move in opposite directions with equal speeds.