When two blocks connected by a spring move towards each other under mutual interaction

Options

A. (a) their velocities are equal
B. (b) their accelerations are equal
C. (c) the force acting on them are equal and opposite
D. (d) All of the above

Answer

To solve the problem of two blocks connected by a spring moving towards each other under mutual interaction, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the System**:
We have two blocks connected by a spring. We will consider the two blocks and the spring as a single system.

2. **Determine Forces Acting on the System**:
Since the blocks are connected by a spring, the force exerted by the spring on each block is an internal force. There are no external forces acting on the system.

3. **Apply Newton's Third Law**:
According to Newton's Third Law, the forces exerted by the spring on each block are equal in magnitude and opposite in direction. If we denote the force exerted by the spring on Block 1 as \( F_{12} \) and on Block 2 as \( F_{21} \), we have:
\[ F_{12} = -F_{21} \]

4. **Consider the Masses of the Blocks**:
Let the masses of the two blocks be \( m_1 \) and \( m_2 \). If the masses are equal (i.e., \( m_1 = m_2 \)), then the accelerations of the blocks will also be equal in magnitude and opposite in direction.

5. **Write the Equations of Motion**:
For Block 1, the equation of motion can be written as:
\[ m_1 a_1 = F_{12} \]
For Block 2, the equation of motion is:
\[ m_2 a_2 = F_{21} \]
Since \( F_{21} = -F_{12} \), we can substitute this into the second equation.

6. **Acceleration Relationship**:
If \( m_1 = m_2 \), then \( a_1 = -a_2 \). This means that the accelerations of the two blocks are equal in magnitude but opposite in direction.

7. **Conclusion**:
The internal forces between the blocks are equal and opposite, and the accelerations of the blocks are also equal and opposite if their masses are equal.

### Final Answer:
The correct conclusion is that the internal forces between the two blocks are equal and opposite, and if the masses are equal, their accelerations will also be equal and opposite.

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