The velocity of a particle of mass 1 kg is given by `v=10sqrtx` the work done by the forceacting on the particle during its motion from `x=4` to `x=9m` is

Options

A. `250`J
B. `300`J
C. `350`J
D. `400`J

Answer

To solve the problem, we need to calculate the work done by the force acting on a particle as it moves from \( x = 4 \) m to \( x = 9 \) m, given that the velocity \( v \) of the particle is defined by the equation \( v = 10\sqrt{x} \).

### Step-by-Step Solution:

1. **Identify the given parameters:**
- Mass of the particle, \( m = 1 \) kg
- Velocity equation, \( v = 10\sqrt{x} \)

2. **Find the acceleration:**
- We know that acceleration \( a \) can be expressed in terms of velocity \( v \) and position \( x \) using the relation:
\[ a = v \frac{dv}{dx} \]
- First, we need to differentiate \( v \) with respect to \( x \):
\[ v = 10\sqrt{x} = 10x^{1/2} \]
- Differentiating \( v \):
\[ \frac{dv}{dx} = 10 \cdot \frac{1}{2}x^{-1/2} = \frac{5}{\sqrt{x}} \]
- Now substituting \( v \) and \( \frac{dv}{dx} \) into the acceleration formula:
\[ a = 10\sqrt{x} \cdot \frac{5}{\sqrt{x}} = 50 \, \text{m/s}^2 \]

3. **Calculate the force acting on the particle:**
- Using Newton's second law:
\[ F = m \cdot a \]
- Substituting the values:
\[ F = 1 \, \text{kg} \cdot 50 \, \text{m/s}^2 = 50 \, \text{N} \]

4. **Determine the displacement:**
- The displacement \( s \) as the particle moves from \( x = 4 \) m to \( x = 9 \) m is:
\[ s = 9 - 4 = 5 \, \text{m} \]

5. **Calculate the work done by the force:**
- Work done \( W \) is given by the formula:
\[ W = F \cdot s \]
- Since the force and displacement are in the same direction:
\[ W = 50 \, \text{N} \cdot 5 \, \text{m} = 250 \, \text{J} \]

### Final Answer:
The work done by the force acting on the particle during its motion from \( x = 4 \) m to \( x = 9 \) m is **250 Joules**.