The displacement 'x' (in meter) of a particle of mass 'm' (in kg) moving in one dimension under the action of a force is released to time 't' (in sec) by `t^2 = sqrt(x) + 2`. The displacement of the particle when its velocity is zero will be.

Answer

To solve the problem step by step, we will analyze the relationship between displacement \( x \) and time \( t \), find the velocity, and determine the displacement when the velocity is zero.

### Step 1: Start with the given equation
The relationship between displacement \( x \) and time \( t \) is given by:
\[ t^2 = \sqrt{x} + 2 \]

### Step 2: Rearrange the equation
We can rearrange the equation to isolate \( \sqrt{x} \):
\[ \sqrt{x} = t^2 - 2 \]

### Step 3: Square both sides to solve for \( x \)
Now, squaring both sides gives:
\[ x = (t^2 - 2)^2 \]

### Step 4: Expand the equation
Expanding the right-hand side:
\[ x = t^4 - 4t^2 + 4 \]

### Step 5: Differentiate to find velocity
The velocity \( v \) is defined as the derivative of displacement with respect to time:
\[ v = \frac{dx}{dt} \]
Differentiating \( x \):
\[ v = \frac{d}{dt}(t^4 - 4t^2 + 4) \]
Using the power rule:
\[ v = 4t^3 - 8t \]

### Step 6: Set velocity to zero
To find when the velocity is zero, we set the equation to zero:
\[ 4t^3 - 8t = 0 \]

### Step 7: Factor the equation
Factoring out \( 4t \):
\[ 4t(t^2 - 2) = 0 \]
This gives us two factors:
1. \( 4t = 0 \) → \( t = 0 \)
2. \( t^2 - 2 = 0 \) → \( t = \pm \sqrt{2} \)

### Step 8: Find corresponding displacements
Now we will find the displacement \( x \) for each value of \( t \).

1. For \( t = 0 \):
\[ x = (0^2 - 2)^2 = (-2)^2 = 4 \text{ meters} \]

2. For \( t = \sqrt{2} \):
\[ x = (\sqrt{2}^2 - 2)^2 = (2 - 2)^2 = 0 \text{ meters} \]

3. For \( t = -\sqrt{2} \):
\[ x = ((-\sqrt{2})^2 - 2)^2 = (2 - 2)^2 = 0 \text{ meters} \]

### Step 9: Conclusion
The displacement of the particle when its velocity is zero can be either:
- \( 4 \text{ meters} \) (when \( t = 0 \))
- \( 0 \text{ meters} \) (when \( t = \sqrt{2} \) or \( t = -\sqrt{2} \))

### Summary of Results
The possible displacements when the velocity is zero are \( 4 \text{ meters} \) and \( 0 \text{ meters} \).

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