A particle starts travelling on a circle with constant tangential acceleration. The angle between velocity vector and acceleration vector, at the moment when particle completes half the circular track. Is

Options

A. `tan ^(-1) (2pi)`
B. `tan^(-1) (pi)`
C. `tan^(-1) (3pi)`
D. zero

Explanation

To solve the problem, we need to analyze the motion of a particle traveling along a circular path with constant tangential acceleration. We will determine the angle between the velocity vector and the acceleration vector when the particle completes half of the circular track.

### Step-by-Step Solution:

1. **Understanding the Motion**:
- The particle is moving in a circular path with a constant tangential acceleration (denoted as \( a_T \)).
- As the particle moves, it experiences two types of acceleration:
- Tangential acceleration (\( a_T \)), which is responsible for changing the speed of the particle along the circular path.
- Centripetal acceleration (\( a_C \)), which is directed towards the center of the circle and is responsible for changing the direction of the velocity vector.

2. **Position at Halfway Point**:
- When the particle completes half of the circular track, it will be at the point directly opposite its starting point on the circle.
- At this point, the velocity vector (\( \vec{v} \)) will be tangent to the circle and directed along the horizontal axis (assuming the starting point is at the top of the circle).

3. **Direction of Accelerations**:
- The tangential acceleration (\( a_T \)) will also be directed along the tangent to the circle at the halfway point, which is in the same direction as the velocity vector.
- The centripetal acceleration (\( a_C \)) will be directed towards the center of the circle.

4. **Angle Between Velocity and Acceleration**:
- Since the tangential acceleration (\( a_T \)) is in the same direction as the velocity vector (\( \vec{v} \)), the angle between them is \( 0^\circ \).
- The centripetal acceleration (\( a_C \)) does not affect the angle between the velocity and tangential acceleration since it is always directed towards the center of the circle.

5. **Conclusion**:
- The angle between the velocity vector and the tangential acceleration vector when the particle completes half the circular track is \( 0^\circ \).

### Final Answer:
The angle between the velocity vector and the acceleration vector at the moment when the particle completes half the circular track is \( 0^\circ \).

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