If y, v and a represent displacement velocity and acceleration at any instant for a particle executing SHM, which of the following statements are true?

Options

A. v and y may have some direction
B. v and a may have same direction
C. a and y may have same direction
D. a and v may have opposite directions.

Answer

To analyze the statements regarding displacement (y), velocity (v), and acceleration (a) for a particle executing Simple Harmonic Motion (SHM), we can summarize the properties of these quantities in SHM and determine the validity of each statement step by step.

### Step-by-Step Solution:

1. **Understanding the Relationship in SHM**:
- In SHM, the displacement (y) from the mean position can be expressed as:
\[ y = A \sin(\omega t + \phi) \]
where \( A \) is the amplitude, \( \omega \) is the angular frequency, and \( \phi \) is the phase constant.

2. **Velocity in SHM**:
- The velocity (v) can be derived by differentiating the displacement with respect to time:
\[ v = \frac{dy}{dt} = A \omega \cos(\omega t + \phi) \]
- The direction of velocity depends on the position of the particle in its oscillation.

3. **Acceleration in SHM**:
- The acceleration (a) is the second derivative of displacement:
\[ a = \frac{d^2y}{dt^2} = -A \omega^2 \sin(\omega t + \phi) = -\omega^2 y \]
- This shows that acceleration is always directed towards the mean position and is opposite to the displacement.

4. **Evaluating the Statements**:
- **Statement 1**: "Velocity and displacement may have the same direction."
- **True**: When the particle is moving away from the mean position, both velocity and displacement are in the same direction.

- **Statement 2**: "Velocity and acceleration may have the same direction."
- **True**: When the particle is moving towards the mean position, the velocity is in the negative direction while the acceleration is also in the negative direction (both directed towards the mean position).

- **Statement 3**: "Acceleration and displacement may have the same direction."
- **False**: Acceleration is always opposite to displacement, as shown by the equation \( a = -\omega^2 y \).

- **Statement 4**: "Acceleration and velocity may have opposite directions."
- **True**: When the particle is at maximum displacement, the velocity is zero, and acceleration is maximum (directed towards the mean position), indicating opposite directions.

5. **Conclusion**:
- The true statements are: 1, 2, and 4.
- The false statement is: 3.

### Final Answer:
- True Statements: 1, 2, and 4.
- False Statement: 3.