The position x of a particle moving along x - axis at time (t) is given by the equation `t=sqrtx+2`, where x is in metres and t in seconds. Find the work done by the force in first four seconds

Options

A. Zero
B. 2 J
C. 4 J
D. 8 J

Answer

To solve the problem, we need to find the work done by the force acting on a particle moving along the x-axis, given the position-time relationship \( t = \sqrt{x} + 2 \). We will follow these steps:

### Step 1: Rearranging the Equation
We start with the equation:
\[ t = \sqrt{x} + 2 \]
Rearranging this gives:
\[ \sqrt{x} = t - 2 \]
Now, squaring both sides, we find:
\[ x = (t - 2)^2 \]

### Step 2: Finding the Velocity
The velocity \( v \) is given by the derivative of position \( x \) with respect to time \( t \):
\[ v = \frac{dx}{dt} \]
Substituting \( x = (t - 2)^2 \):
\[ v = \frac{d}{dt}((t - 2)^2) = 2(t - 2) \cdot \frac{d(t - 2)}{dt} = 2(t - 2) \]
Thus, the velocity is:
\[ v = 2t - 4 \quad \text{(in meters per second)} \]

### Step 3: Finding the Acceleration
The acceleration \( a \) is given by the derivative of velocity \( v \) with respect to time \( t \):
\[ a = \frac{dv}{dt} \]
Substituting \( v = 2t - 4 \):
\[ a = \frac{d}{dt}(2t - 4) = 2 \quad \text{(in meters per second squared)} \]

### Step 4: Finding the Force
Using Newton's second law, the force \( F \) is given by:
\[ F = m \cdot a \]
Assuming the mass of the particle is \( m \):
\[ F = 2m \quad \text{(in Newtons)} \]

### Step 5: Finding Displacement at \( t = 0 \) and \( t = 4 \)
Now we calculate the displacement at \( t = 0 \):
\[ x_0 = (0 - 2)^2 = 4 \quad \text{(in meters)} \]
Next, we calculate the displacement at \( t = 4 \):
\[ x_4 = (4 - 2)^2 = 4 \quad \text{(in meters)} \]

### Step 6: Finding the Change in Displacement
The change in displacement \( \Delta x \) is:
\[ \Delta x = x_4 - x_0 = 4 - 4 = 0 \quad \text{(in meters)} \]

### Step 7: Calculating Work Done
The work done \( W \) by the force is given by:
\[ W = F \cdot \Delta x \]
Substituting the values:
\[ W = 2m \cdot 0 = 0 \quad \text{(in Joules)} \]

### Conclusion
The work done by the force in the first four seconds is:
\[ \boxed{0 \text{ Joules}} \]