Assertion : In ` x = A cos omega t`, the dot product of acceleration and velocity is positive for time interval `0 lt t lt (pi)/(2omega)`. Reason : Angle between them is `0^(@)` .

Options

A. If both Assertion and Reason are true and the Reason is correct explanation of the Assertion.
B. If both Assertion and Reason are true but Reason is not the correct explanation of Assertion.
C. If Assertion is true, but the Reason is false.
D. If Assertion is false but the Reason is true.

Explanation

To solve the problem, we need to analyze the assertion and reason given in the question step by step.

### Step 1: Understand the Assertion
The assertion states that in the equation \( x = A \cos(\omega t) \), the dot product of acceleration and velocity is positive for the time interval \( 0 < t < \frac{\pi}{2\omega} \).

### Step 2: Find the Expressions for Velocity and Acceleration
1. **Position**: Given \( x = A \cos(\omega t) \).
2. **Velocity**: The velocity \( v \) is the derivative of position with respect to time:
\[ v = \frac{dx}{dt} = -A \omega \sin(\omega t) \]
3. **Acceleration**: The acceleration \( a \) is the derivative of velocity with respect to time:
\[ a = \frac{dv}{dt} = -A \omega^2 \cos(\omega t) \]

### Step 3: Analyze the Time Interval
For the time interval \( 0 < t < \frac{\pi}{2\omega} \):
- At \( t = 0 \), \( \sin(\omega t) = 0 \) and \( \cos(\omega t) = 1 \).
- As \( t \) increases towards \( \frac{\pi}{2\omega} \), \( \sin(\omega t) \) increases from \( 0 \) to \( 1 \) and \( \cos(\omega t) \) decreases from \( 1 \) to \( 0 \).

### Step 4: Determine the Signs of Velocity and Acceleration
- **Velocity**: Since \( v = -A \omega \sin(\omega t) \), for \( 0 < t < \frac{\pi}{2\omega} \), \( v \) is negative (as \( \sin(\omega t) \) is positive).
- **Acceleration**: Since \( a = -A \omega^2 \cos(\omega t) \), for \( 0 < t < \frac{\pi}{2\omega} \), \( a \) is negative (as \( \cos(\omega t) \) is positive).

### Step 5: Calculate the Dot Product
The dot product of acceleration and velocity is given by:
\[ \vec{a} \cdot \vec{v} = |\vec{a}| |\vec{v}| \cos(\theta) \]
Where \( \theta \) is the angle between \( \vec{a} \) and \( \vec{v} \). Since both \( \vec{a} \) and \( \vec{v} \) are negative in this interval, the angle \( \theta \) between them is \( 0^\circ \) (they are in the same direction).

Thus, the dot product becomes:
\[ \vec{a} \cdot \vec{v} = |\vec{a}| |\vec{v}| \cos(0^\circ) = |\vec{a}| |\vec{v}| \]
Since both magnitudes are positive, the dot product is positive.

### Step 6: Conclusion on the Assertion
The assertion is true because the dot product of acceleration and velocity is indeed positive for the specified time interval.

### Step 7: Analyze the Reason
The reason states that the angle between them is \( 0^\circ \). This is correct because both acceleration and velocity are in the same direction (both are negative).

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