If v is the velocity of a body moving along x-axis, then acceleration of body is

Options

A. `(dv)/(dx)`
B. `v(dv)/(dx)`
C. `x(du)/(dx)`
D. `v(dx)/(dv)`

Answer

To solve the problem of finding the acceleration of a body moving along the x-axis when given its velocity \( v \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Definitions**:
- Velocity \( v \) is defined as the rate of change of displacement \( x \) with respect to time \( t \):
\[ v = \frac{dx}{dt} \]
- Acceleration \( a \) is defined as the rate of change of velocity \( v \) with respect to time \( t \):
\[ a = \frac{dv}{dt} \]

2. **Relate Time to Displacement**:
- We can express \( dt \) in terms of \( dx \) and \( v \):
\[ dt = \frac{dx}{v} \]

3. **Substitute \( dt \) into the Acceleration Formula**:
- Now, we substitute \( dt \) in the acceleration formula:
\[ a = \frac{dv}{dt} = \frac{dv}{dx} \cdot \frac{dx}{dt} \]
- Since \( \frac{dx}{dt} = v \), we can rewrite the equation as:
\[ a = \frac{dv}{dx} \cdot v \]

4. **Final Expression for Acceleration**:
- Thus, the expression for acceleration becomes:
\[ a = v \frac{dv}{dx} \]

5. **Identify the Correct Option**:
- From the options given, the correct expression for the acceleration of the body is:
\[ a = v \frac{dv}{dx} \]
- Therefore, the correct answer is option 2: \( v \frac{dv}{dx} \).