A particle of mass 2kg travels along a straight line with velocity `v=asqrtx`, where a is a constant. The work done by net force during the displacement of particle from `x=0` to `x=4m` is

Options

A. `a^(2)`
B. `2a^(2)`
C. `4a^(2)`
D. `sqrt2a^(2)`

Answer

To solve the problem step by step, we need to find the work done by the net force on a particle of mass 2 kg as it travels from \( x = 0 \) to \( x = 4 \) m, given that its velocity is \( v = a \sqrt{x} \), where \( a \) is a constant.

### Step 1: Identify the given information
- Mass of the particle, \( m = 2 \) kg
- Velocity of the particle, \( v = a \sqrt{x} \)

### Step 2: Find the expression for acceleration
Acceleration \( a \) can be expressed as the derivative of velocity with respect to time, \( a = \frac{dv}{dt} \). We can also express it using the chain rule as:
\[ a = \frac{dv}{dx} \cdot \frac{dx}{dt} \]
Here, \( \frac{dx}{dt} = v \).

### Step 3: Calculate \( \frac{dv}{dx} \)
Given \( v = a \sqrt{x} \), we differentiate with respect to \( x \):
\[ \frac{dv}{dx} = \frac{d}{dx}(a \sqrt{x}) = a \cdot \frac{1}{2\sqrt{x}} = \frac{a}{2\sqrt{x}} \]

### Step 4: Substitute \( \frac{dv}{dx} \) into the acceleration formula
Now substituting \( \frac{dv}{dx} \) into the acceleration equation:
\[ a = \frac{dv}{dx} \cdot v = \left(\frac{a}{2\sqrt{x}}\right) \cdot (a \sqrt{x}) = \frac{a^2}{2} \]

### Step 5: Calculate the net force
Using Newton's second law, the net force \( F \) is given by:
\[ F = m \cdot a = 2 \cdot \frac{a^2}{2} = a^2 \]

### Step 6: Calculate the work done
The work done \( W \) by the net force as the particle moves from \( x = 0 \) to \( x = 4 \) m is given by:
\[ W = \int_{0}^{4} F \, dx = \int_{0}^{4} a^2 \, dx \]
Since \( a^2 \) is a constant, we can take it out of the integral:
\[ W = a^2 \int_{0}^{4} dx = a^2 [x]_{0}^{4} = a^2 (4 - 0) = 4a^2 \]

### Final Answer
The work done by the net force during the displacement of the particle from \( x = 0 \) to \( x = 4 \) m is:
\[ W = 4a^2 \]