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 of a particle at rest disintegrating into two particles of equal mass, we will follow these steps:

### Step 1: Understand the Initial Conditions
The particle is initially at rest, which means its initial momentum is zero.

**Hint:** Remember that momentum is the product of mass and velocity, and if the velocity is zero, the momentum is also zero.

### Step 2: Define the Masses
Let the mass of the original particle be \(2m\). After disintegration, it breaks into two fragments, each of mass \(m\).

**Hint:** When dealing with conservation of momentum, it’s important to clearly define the masses involved.

### Step 3: Apply Conservation of Momentum
According to the law of conservation of momentum, the total momentum before disintegration must equal the total momentum after disintegration.

Since the initial momentum is zero, we have:
\[ p_{\text{initial}} = 0 = p_{\text{final}} \]

**Hint:** Conservation of momentum states that in an isolated system, the total momentum remains constant if no external forces act on it.

### Step 4: Set Up the Equation for Final Momentum
Let’s assume that one fragment moves to the left with velocity \(v_1\) and the other fragment moves to the right with velocity \(v_2\). The equation for final momentum can be expressed as:
\[ mv_1 + mv_2 = 0 \]

**Hint:** The direction of the velocities matters; one will be negative (left) and the other positive (right).

### Step 5: Simplify the Equation
Dividing the entire equation by \(m\) (since \(m \neq 0\)), we get:
\[ v_1 + v_2 = 0 \]
This implies:
\[ v_1 = -v_2 \]

**Hint:** The negative sign indicates that the two velocities are in opposite directions.

### Step 6: Conclude the Relationship Between Velocities
From the equation \(v_1 = -v_2\), we can conclude that the magnitudes of the velocities are equal:
\[ |v_1| = |v_2| \]

**Hint:** Equal magnitudes with opposite signs indicate that the particles move away from each other.

### Final Conclusion
Thus, we conclude that the two fragments will move in opposite directions with equal velocities.

**Final Answer:** The two fragments will move in opposite directions with equal velocity.