Let `v` and `a` be the instantaneous velocity and acceleration of a particle moving in a plane. Then rate of change of speed `(dv)/(dt)` of the particle is equal to

Options

A. `|a|`
B. `(v.a)/v`
C. the component of `a` parallel to `v`
D. the component of `a` perpendicular to `v`

Answer

To find the rate of change of speed \( \frac{dv}{dt} \) of a particle moving in a plane, we can follow these steps:

### Step 1: Understand the Definitions
- **Instantaneous Velocity (\( v \))**: The velocity of the particle at a specific instant.
- **Instantaneous Acceleration (\( a \))**: The acceleration of the particle at a specific instant.

### Step 2: Identify the Rate of Change of Speed
The rate of change of speed is defined as:
\[ \frac{dv}{dt} \]
This represents how quickly the speed of the particle is changing over time.

### Step 3: Relate Speed Change to Acceleration
The rate of change of speed is associated with the tangential component of acceleration (\( a_t \)), which is the component of acceleration that acts in the direction of the velocity. Therefore, we can express this as:
\[ \frac{dv}{dt} = a_t \]

### Step 4: Resolve Acceleration into Components
The total acceleration \( a \) can be resolved into two components:
1. **Tangential Acceleration (\( a_t \))**: This is the component of acceleration in the direction of the velocity.
2. **Normal Acceleration (\( a_n \))**: This is the component of acceleration perpendicular to the velocity.

### Step 5: Calculate Tangential Acceleration
The tangential acceleration can be calculated using the angle \( \theta \) between the velocity vector \( v \) and the acceleration vector \( a \):
\[ a_t = a \cos(\theta) \]
Alternatively, using the dot product:
\[ a_t = \frac{a \cdot v}{|v|} \]
where \( a \cdot v \) is the dot product of the acceleration and velocity vectors.

### Step 6: Conclusion
Thus, we can conclude that the rate of change of speed \( \frac{dv}{dt} \) is equal to the tangential component of acceleration, which can be expressed as:
\[ \frac{dv}{dt} = a_t = \frac{a \cdot v}{|v|} \]

### Final Answer
The rate of change of speed \( \frac{dv}{dt} \) of the particle is equal to the component of acceleration parallel to the velocity, or mathematically:
\[ \frac{dv}{dt} = a_t = \frac{a \cdot v}{|v|} \]