Assertion: In simple harmonic motion the velocity is maximum when the acceleration is minimum
Reason : Displacement and velocity of `SHM`differ in phase by `(pi)/(2)`

Options

A. If both Assertion & Reason are true and the reason is the correct explanation of the assertion, then mark (1)
B. If both Assertion & Reason are true but the reason is not the correct explanation of the assertion, then mark (2)
C. If Assertion is true statement but Reason is false then mark (3)
D. If both Assertion and Reason are false statements, then mark (4)

Answer

To solve the assertion-reason question, we will analyze both the assertion and the reason step by step.

### Step 1: Understanding the Assertion
The assertion states: "In simple harmonic motion, the velocity is maximum when the acceleration is minimum."

1. In simple harmonic motion (SHM), the acceleration (a) is given by the equation:
\[ a = -\omega^2 x \]
where \(x\) is the displacement from the mean position and \(\omega\) is the angular frequency.

2. When the displacement \(x\) is maximum (i.e., at the extremes of the motion), the acceleration \(a\) is also maximum (in magnitude). Conversely, when \(x\) is zero (the mean position), the acceleration is zero.

3. The velocity \(v\) in SHM is given by:
\[ v = \omega \sqrt{A^2 - x^2} \]
where \(A\) is the amplitude of the motion. The velocity is maximum when \(x = 0\).

4. Therefore, at the mean position (where \(x = 0\)), the velocity is maximum, and at this point, the acceleration is minimum (zero). Hence, the assertion is true.

### Step 2: Understanding the Reason
The reason states: "Displacement and velocity of SHM differ in phase by \(\frac{\pi}{2}\)."

1. The displacement in SHM can be expressed as:
\[ x(t) = A \cos(\omega t) \]

2. The velocity is the time derivative of displacement:
\[ v(t) = \frac{dx}{dt} = -A \omega \sin(\omega t) \]

3. The functions \(\cos(\omega t)\) and \(-\sin(\omega t)\) are indeed out of phase by \(\frac{\pi}{2}\) radians. This means when displacement is at a maximum or minimum, the velocity is zero, and when the displacement is zero, the velocity is at its maximum.

4. Thus, the reason is also true.

### Conclusion
Both the assertion and the reason are true, and the reason correctly explains the assertion.

### Final Answer
Both the assertion and the reason are true, and the reason is the correct explanation for the assertion.

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