if `theta` is angle between the velocity and acceleration of a particle moving on a circular path with decreasing speed, then

Options

A. `theta = 90^(@)`
B. `0^(@) lt theta lt 90^(@)`
C. `90^(@) lt theta lt 180^(@)`
D. `0^(@) le theta le 180^(@)`

Answer

To solve the problem, we need to analyze the motion of a particle moving in a circular path with decreasing speed. We will determine the angle \( \theta \) between the velocity vector and the acceleration vector.

### Step-by-Step Solution:

1. **Understanding the Motion**:
- The particle is moving in a circular path, which means it has centripetal acceleration directed towards the center of the circle.
- Since the speed of the particle is decreasing, there is also a tangential acceleration acting in the direction opposite to the velocity.

2. **Identifying the Components of Acceleration**:
- The total acceleration \( \vec{a} \) of the particle can be broken down into two components:
- **Centripetal Acceleration (\( \vec{a}_c \))**: This is directed towards the center of the circular path.
- **Tangential Acceleration (\( \vec{a}_t \))**: This is directed opposite to the velocity vector since the speed is decreasing.

3. **Direction of Velocity and Acceleration**:
- Let \( \vec{v} \) be the velocity vector, which is tangent to the circular path.
- The centripetal acceleration \( \vec{a}_c \) is perpendicular to \( \vec{v} \) (90 degrees).
- The tangential acceleration \( \vec{a}_t \) is in the opposite direction of \( \vec{v} \).

4. **Resultant Acceleration**:
- The resultant acceleration \( \vec{a} \) is the vector sum of \( \vec{a}_c \) and \( \vec{a}_t \).
- Since \( \vec{a}_c \) is perpendicular to \( \vec{v} \) and \( \vec{a}_t \) is opposite to \( \vec{v} \), the angle \( \theta \) between the velocity vector \( \vec{v} \) and the resultant acceleration \( \vec{a} \) can be determined.

5. **Finding the Angle \( \theta \)**:
- The angle between the velocity vector \( \vec{v} \) and the centripetal acceleration \( \vec{a}_c \) is 90 degrees.
- The angle \( \theta \) between the velocity vector \( \vec{v} \) and the resultant acceleration \( \vec{a} \) must be less than 90 degrees (because \( \vec{a}_t \) pulls the resultant acceleration towards the opposite direction of \( \vec{v} \)).
- Therefore, \( \theta \) must be between 90 degrees and 180 degrees.

### Conclusion:
The angle \( \theta \) between the velocity and acceleration of a particle moving on a circular path with decreasing speed is between 90 degrees and 180 degrees.