Assertion: In S.H.M., the velocity and displacement of the particle are in the same phase.
Reason: Velocity is the ratio of displacement to the time taken.

Options

A. Assertion is True, Reason is True, Reason is a correct explanation for Assertion
B. Assertion is True, Reason is True, Reason is not a correct explanation for Assertion
C. Assertion is True, Reason is False
D. Assertion is False but, Reason is True.

Answer

To solve the assertion-reason question, we need to analyze both the assertion and the reason provided.

### Step-by-Step Solution:

1. **Understanding the Assertion**:
The assertion states that in Simple Harmonic Motion (S.H.M.), the velocity and displacement of the particle are in the same phase.

2. **Displacement in S.H.M.**:
In S.H.M., the displacement \( x \) of a particle can be expressed as:
\[ x(t) = A \sin(\omega t) \]
where \( A \) is the amplitude and \( \omega \) is the angular frequency.

3. **Calculating Velocity**:
The velocity \( v \) of the particle is the time derivative of displacement:
\[ v(t) = \frac{dx}{dt} = \frac{d}{dt}(A \sin(\omega t)) = A \omega \cos(\omega t) \]

4. **Phase Relationship**:
The displacement \( x(t) = A \sin(\omega t) \) is in the sine function, while the velocity \( v(t) = A \omega \cos(\omega t) \) is in the cosine function.

5. **Phase Difference**:
The cosine function can be expressed in terms of sine:
\[ \cos(\omega t) = \sin\left(\omega t + \frac{\pi}{2}\right) \]
This indicates that the velocity is \( \frac{\pi}{2} \) radians (or 90 degrees) ahead of the displacement in phase.

6. **Conclusion on the Assertion**:
Since the phase difference between displacement and velocity is \( \frac{\pi}{2} \), they are not in the same phase. Therefore, the assertion is **false**.

7. **Understanding the Reason**:
The reason states that velocity is the ratio of displacement to the time taken. This is indeed true, as velocity \( v \) is defined as:
\[ v = \frac{dx}{dt} \]
This statement is correct, but it does not support the assertion.

8. **Final Conclusion**:
The assertion is false, and the reason is true. Thus, the correct answer is that the assertion is false and the reason is true.

### Summary:
- Assertion: False
- Reason: True