The speed of a particle moving in a circle is increasing. The dot product of its acceleration and velocity is

Options

A. negative
B. zero
C. positive
D. may be positive or negative

Explanation

To solve the problem, we need to analyze the motion of a particle moving in a circular path with increasing speed. We will determine the dot product of its acceleration and velocity.

### Step-by-Step Solution:

1. **Understanding the Components of Motion:**
- When a particle moves in a circular path, its motion can be described by two components of acceleration:
- **Tangential Acceleration (a_t)**: This is responsible for the change in the speed of the particle along the circular path.
- **Centripetal Acceleration (a_c)**: This is directed towards the center of the circle and is responsible for changing the direction of the velocity.

2. **Given Information:**
- The problem states that the speed of the particle is increasing. This implies that the tangential acceleration (a_t) is positive.

3. **Velocity and Acceleration Vectors:**
- Let **v** be the velocity vector of the particle. Since the particle is moving in a circle, the direction of **v** is tangent to the circular path.
- Let **a** be the total acceleration vector, which can be expressed as the vector sum of tangential acceleration and centripetal acceleration:
\[ \mathbf{a} = \mathbf{a_t} + \mathbf{a_c} \]

4. **Dot Product of Acceleration and Velocity:**
- The dot product of two vectors **a** and **v** is given by:
\[ \mathbf{a} \cdot \mathbf{v} = |\mathbf{a}| |\mathbf{v}| \cos(\theta) \]
- Here, **θ** is the angle between the acceleration vector and the velocity vector.

5. **Analyzing the Angle:**
- Since the tangential acceleration (which is positive) acts in the same direction as the velocity vector (because it increases the speed), the angle between the tangential acceleration and the velocity vector is 0 degrees.
- The centripetal acceleration is perpendicular to the velocity vector (90 degrees), contributing nothing to the dot product.

6. **Calculating the Dot Product:**
- The dot product can be simplified as:
\[ \mathbf{a} \cdot \mathbf{v} = \mathbf{a_t} \cdot \mathbf{v} + \mathbf{a_c} \cdot \mathbf{v} \]
- Since \(\mathbf{a_c} \cdot \mathbf{v} = 0\) (because they are perpendicular), we have:
\[ \mathbf{a} \cdot \mathbf{v} = \mathbf{a_t} \cdot \mathbf{v} \]
- Since \(\mathbf{a_t}\) is in the same direction as \(\mathbf{v}\), we conclude:
\[ \mathbf{a_t} \cdot \mathbf{v} > 0 \]

7. **Final Conclusion:**
- Therefore, the dot product of the acceleration and velocity is positive.

### Final Answer:
The dot product of the acceleration and velocity is positive.