Two particles are performing SHM in same phase. It means that

Options

A. the two partcles must have same distance from the mean posotion simultaneously
B. two particle may have same distance from the mean position simultaneously
C. the two particles must have maximum speed simultaneously
D. the two particles may have maximum speed simultaneously

Explanation

To solve the question regarding two particles performing Simple Harmonic Motion (SHM) in the same phase, we will analyze the implications of being in the same phase and what it means for their motion.

### Step-by-Step Solution:

1. **Understanding SHM and Phase**:
- In SHM, the position of a particle can be described by the equation:
\[ x(t) = A \sin(\omega t + \phi) \]
where \(A\) is the amplitude, \(\omega\) is the angular frequency, and \(\phi\) is the phase constant.
- When two particles are said to be in the same phase, it means that at any given time \(t\), their phase angles are equal. This can be expressed as:
\[ \phi_1 = \phi_2 \]

2. **Same Distance from Mean Position**:
- If two particles are in the same phase, they will reach the same position \(x\) at the same time \(t\) if their amplitudes are equal. Thus, they may have the same distance from the mean position simultaneously, but this is not guaranteed unless their amplitudes are also equal.
- Therefore, the statement "the two particles must have the same distance from the mean position simultaneously" is incorrect.

3. **Same Speed**:
- The speed of a particle in SHM is given by:
\[ v(t) = \frac{dx}{dt} = A\omega \cos(\omega t + \phi) \]
- Since both particles are in the same phase, they will have the same value of \(\cos(\omega t + \phi)\) at any time \(t\). This means they will also have the same speed at that time.
- Hence, the statement "the two particles must have the same speed simultaneously" is correct.

4. **Maximum Speed**:
- The maximum speed of a particle in SHM occurs when it passes through the mean position. Since both particles are in the same phase, they will reach their maximum speed at the same time.
- Thus, the statement "the two particles may have maximum speeds simultaneously" is also correct.

### Conclusion:
- The correct interpretations of the statements are:
- **B**: Two particles must have the same distance from the mean position simultaneously - **Incorrect**.
- **C**: Two particles must have the same speed simultaneously - **Correct**.
- **D**: Two particles may have maximum speeds simultaneously - **Correct**.

Therefore, the final answer is that statements C and D are correct.