Two blocks are connected to the two free ends of a spring of spring constant K . Both the blocks are moved apart to extend the psring beyond its natural length and then released . In subsequent motion

Options

A. Accelerations of both the objects are equal and opposite of each other .
B. Velocities of both the bjects are equal and opposite of each other .
C. Forces acting on both the objects are equal and opposite of each other
D. Linear momenta of both the objects are equal and opposite of each other .

Explanation

To solve the problem, let's analyze the situation step by step:

### Step 1: Understand the System
We have two blocks connected by a spring. When the blocks are pulled apart and released, they will experience forces due to the spring trying to return to its natural length.

### Step 2: Identify Forces Acting on the Blocks
When the spring is extended, it exerts a force on both blocks. The force exerted by the spring on each block is given by Hooke's Law:
\[ F = -kx \]
where \( k \) is the spring constant and \( x \) is the displacement from the natural length of the spring.

### Step 3: Analyze the Acceleration
Let’s denote the masses of the blocks as \( m_1 \) and \( m_2 \). The acceleration of each block can be calculated using Newton's second law:
\[ a_1 = \frac{F_{spring}}{m_1} \quad \text{and} \quad a_2 = \frac{F_{spring}}{m_2} \]
Since the forces acting on both blocks are equal in magnitude but opposite in direction, the accelerations \( a_1 \) and \( a_2 \) will not necessarily be equal unless the masses are the same.

### Step 4: Analyze the Velocities
Since there are no external forces acting on the system, the total momentum of the system is conserved. Initially, the system is at rest, so the total momentum is zero:
\[ 0 = m_1 v_1 + m_2 v_2 \]
This implies:
\[ m_1 v_1 = -m_2 v_2 \]
This means that the velocities \( v_1 \) and \( v_2 \) are not equal but are opposite in direction.

### Step 5: Analyze the Forces
As discussed, the forces acting on both blocks are equal in magnitude and opposite in direction:
\[ F_{1} = kx \quad \text{and} \quad F_{2} = -kx \]
Thus, the forces are equal and opposite, confirming that option C is correct.

### Step 6: Analyze the Linear Momentum
From the conservation of momentum, we have already established that:
\[ m_1 v_1 + m_2 v_2 = 0 \]
This indicates that the linear momentum of the two blocks is equal in magnitude and opposite in direction, confirming that option D is also correct.

### Conclusion
The correct answers to the options provided in the question are:
- C: Forces acting on both objects are equal and opposite to each other.
- D: Linear momentum of both objects are equal and opposite to each other.

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