Assertion:- A particle is moving in a circle with constant tangential acceleration such that its speed v is increasing. Angle made by resultant acceleration of the particle with tangential acceleration increases with time.
Reason:- Tangential acceleration `=|(dvecv)/(dt)|` and centripetal acceleration `=(v^(2))/(R)`

Options

A. If both Assertion `&` Reason are True `&` the Reason is a correct explanation of the Assertion.
B. If both Assertion `&` Reason are True but Reason is not a correct explanation of the Assertion.
C. If Assertion is True but the Reason is False.
D. If both Assertion `&` Reason are False.

Answer

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

### Step-by-Step Solution:

1. **Understanding the Assertion**:
The assertion states that a particle is moving in a circle with constant tangential acceleration, and its speed \( v \) is increasing. It claims that the angle made by the resultant acceleration of the particle with the tangential acceleration increases with time.

2. **Identifying the Components of Acceleration**:
- The **tangential acceleration** \( a_t \) is responsible for the change in the speed of the particle along the circular path. It is constant in this case.
- The **centripetal acceleration** \( a_c \) is given by the formula \( a_c = \frac{v^2}{R} \), where \( R \) is the radius of the circular path. This acceleration is directed towards the center of the circle and depends on the speed \( v \).

3. **Resultant Acceleration**:
The resultant acceleration \( a_r \) can be found using the vector sum of tangential and centripetal accelerations. The angle \( \theta \) between the resultant acceleration and the tangential acceleration can be expressed using the tangent function:
\[ \tan(\theta) = \frac{a_c}{a_t} = \frac{\frac{v^2}{R}}{a_t} \]

4. **Analyzing the Change in Angle**:
Since the tangential acceleration \( a_t \) is constant and the speed \( v \) is increasing, the centripetal acceleration \( a_c \) will also increase as \( v \) increases. Therefore, as \( v \) increases, \( a_c \) increases, leading to an increase in \( \tan(\theta) \). Consequently, the angle \( \theta \) will also increase with time.

5. **Conclusion for Assertion**:
The assertion is true because the angle made by the resultant acceleration with the tangential acceleration indeed increases with time due to the increasing speed of the particle.

6. **Understanding the Reason**:
The reason states that tangential acceleration is defined as \( |(d\vec{v})/(dt)| \) and centripetal acceleration is \( \frac{v^2}{R} \). This is a correct statement regarding the definitions of tangential and centripetal accelerations.

7. **Evaluating the Truth of the Reason**:
Although the reason is true, it does not directly support the assertion about the angle increasing with time. Thus, while both the assertion and reason are true, the reason does not provide a justification for the assertion.

### Final Evaluation:
- **Assertion**: True
- **Reason**: True
- **Conclusion**: The assertion is true, and the reason is true, but the reason does not explain the assertion.