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 find the acceleration of a body moving along the x-axis when its velocity is given as \( v \), we can follow these steps:

### Step-by-Step Solution:

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

2. **Apply the Chain Rule**:
- To express acceleration in terms of displacement, we can use the chain rule of differentiation. We can write:
\[ a = \frac{dv}{dt} = \frac{dv}{dx} \cdot \frac{dx}{dt} \]
- Here, \( \frac{dx}{dt} \) is the velocity \( v \).

3. **Substitute the Velocity**:
- Substitute \( v \) for \( \frac{dx}{dt} \) in the equation:
\[ a = \frac{dv}{dx} \cdot v \]

4. **Final Expression for Acceleration**:
- Therefore, the acceleration of the body can be expressed as:
\[ a = v \cdot \frac{dv}{dx} \]

### Final Answer:
The acceleration of the body moving along the x-axis is given by:
\[ a = v \cdot \frac{dv}{dx} \]

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