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 moves from \( x = 0 \) to \( x = 4 \, m \) with a velocity given by \( v = a \sqrt{x} \), where \( a \) is a constant.

### Step 1: Find the expression for acceleration
The velocity \( v \) is given as:
\[ v = a \sqrt{x} \]
To find acceleration \( a \), we can use the relationship:
\[ a = \frac{dv}{dt} = \frac{dv}{dx} \cdot \frac{dx}{dt} = v \frac{dv}{dx} \]
Substituting \( v \) into the equation:
\[ \frac{dv}{dx} = \frac{d}{dx}(a \sqrt{x}) = a \cdot \frac{1}{2\sqrt{x}} = \frac{a}{2\sqrt{x}} \]
Thus, we have:
\[ a = v \cdot \frac{dv}{dx} = (a \sqrt{x}) \cdot \left(\frac{a}{2\sqrt{x}}\right) = \frac{a^2}{2} \]

### Step 2: Find the net force acting on the particle
Using Newton's second law, the net force \( F \) can be calculated as:
\[ F = m \cdot a \]
Given that the mass \( m = 2 \, kg \) and \( a = \frac{a^2}{2} \):
\[ F = 2 \cdot \frac{a^2}{2} = a^2 \]

### Step 3: Calculate the work done by the net force
The work done \( W \) by the net force during the displacement from \( x = 0 \) to \( x = 4 \, m \) is given by:
\[ W = \int F \, dx \]
Since \( F = a^2 \) is a constant, the work done can be simplified to:
\[ W = F \cdot \Delta x = a^2 \cdot (4 - 0) = 4a^2 \]

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