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 step by step, we will follow these steps:

### Step 1: Rearranging the given equation
The position \( x \) of a particle is given by the equation:
\[ t = \sqrt{x} + 2 \]
We can rearrange this equation to express \( x \) in terms of \( t \):
\[ t - 2 = \sqrt{x} \]
Now, squaring both sides:
\[ (t - 2)^2 = x \]
Thus, we have:
\[ x = (t - 2)^2 \]

### Step 2: Finding the velocity
To find the velocity \( v \), we need to differentiate the position \( x \) with respect to time \( t \):
\[ v = \frac{dx}{dt} = \frac{d}{dt}[(t - 2)^2] \]
Using the chain rule:
\[ v = 2(t - 2) \cdot \frac{d}{dt}(t) = 2(t - 2) \]

### Step 3: Finding initial and final velocities
Now we will calculate the initial and final velocities at \( t = 0 \) seconds and \( t = 4 \) seconds.

- **Initial velocity** at \( t = 0 \):
\[ v_{\text{initial}} = 2(0 - 2) = 2 \times (-2) = -4 \, \text{m/s} \]

- **Final velocity** at \( t = 4 \):
\[ v_{\text{final}} = 2(4 - 2) = 2 \times 2 = 4 \, \text{m/s} \]

### Step 4: Calculating the initial and final kinetic energy
The kinetic energy \( KE \) is given by the formula:
\[ KE = \frac{1}{2} mv^2 \]
Assuming the mass \( m \) of the particle is \( m \) (we will keep it as a variable since it is not provided):

- **Initial kinetic energy**:
\[ KE_{\text{initial}} = \frac{1}{2} m (-4)^2 = \frac{1}{2} m \cdot 16 = 8m \, \text{J} \]

- **Final kinetic energy**:
\[ KE_{\text{final}} = \frac{1}{2} m (4)^2 = \frac{1}{2} m \cdot 16 = 8m \, \text{J} \]

### Step 5: Calculating the work done
According to the work-energy theorem, the work done \( W \) is equal to the change in kinetic energy:
\[ W = KE_{\text{final}} - KE_{\text{initial}} = 8m - 8m = 0 \, \text{J} \]

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