The displacement of a body from a reference point is given by `sqrt(x)=2t-3`, where `'x'` is in metres and it is non negative number, `t` in seconds. This shows that the body `:`

Options

A. is at rest at `t=3//2`
B. is speeding up for `tgt3//2`
C. is retarded for `tlt3//2`
D. is in uniform motion

Explanation

To solve the problem, we start with the given equation for displacement:

\[ \sqrt{x} = 2t - 3 \]

### Step 1: Rearranging the equation
To express displacement \(x\) in terms of time \(t\), we square both sides of the equation:

\[ x = (2t - 3)^2 \]

### Step 2: Finding Velocity
Velocity \(v\) is defined as the rate of change of displacement with respect to time. We differentiate \(x\) with respect to \(t\):

\[ v = \frac{dx}{dt} = \frac{d}{dt}[(2t - 3)^2] \]

Using the chain rule, we have:

\[ v = 2(2t - 3) \cdot \frac{d}{dt}(2t - 3) = 2(2t - 3) \cdot 2 = 4(2t - 3) \]

### Step 3: Finding when the body is at rest
The body is at rest when the velocity \(v = 0\):

\[ 4(2t - 3) = 0 \]

Solving for \(t\):

\[ 2t - 3 = 0 \implies 2t = 3 \implies t = \frac{3}{2} \text{ seconds} \]

### Step 4: Analyzing motion for \(t > \frac{3}{2}\)
For \(t > \frac{3}{2}\):

- The term \(2t - 3\) becomes positive, thus \(v > 0\).
- To check acceleration, we differentiate velocity:

\[ a = \frac{dv}{dt} = \frac{d}{dt}[4(2t - 3)] = 4 \cdot 2 = 8 \text{ m/s}^2 \]

Since both velocity and acceleration are positive, the body is speeding up.

### Step 5: Analyzing motion for \(t < \frac{3}{2}\)
For \(t < \frac{3}{2}\):

- The term \(2t - 3\) becomes negative, thus \(v < 0\).
- The acceleration remains \(8 \text{ m/s}^2\) (positive).

Since velocity is negative and acceleration is positive, the body is retarded.

### Step 6: Checking for uniform motion
Since acceleration is constant (8 m/s²) throughout the motion, the body is in uniform motion.

### Conclusion
Based on the analysis:
1. The body is at rest at \(t = \frac{3}{2}\) seconds.
2. The body is speeding up for \(t > \frac{3}{2}\).
3. The body is retarded for \(t < \frac{3}{2}\).
4. The body is in uniform motion due to constant acceleration.

Thus, all options provided in the question are correct.

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