The displacement of a body is given by `2 s= g t^(2)` where g is a constant. The velocity of the body at any time t is :

Options

A. `g t`
B. `g t//2`
C. `g t^(2)//2`
D. `g t^(3)//6`

Explanation

To find the velocity of the body at any time \( t \), we start with the given displacement equation:

\[ 2s = gt^2 \]

### Step 1: Differentiate the displacement with respect to time
To find the velocity, we need to differentiate the displacement \( s \) with respect to time \( t \). The velocity \( v \) is defined as the rate of change of displacement with respect to time, which can be expressed mathematically as:

\[ v = \frac{ds}{dt} \]

### Step 2: Rewrite the displacement equation
From the given equation, we can express \( s \) in terms of \( t \):

\[ s = \frac{gt^2}{2} \]

### Step 3: Differentiate \( s \) with respect to \( t \)
Now we differentiate \( s \):

\[ \frac{ds}{dt} = \frac{d}{dt}\left(\frac{gt^2}{2}\right) \]

Using the power rule of differentiation, we get:

\[ \frac{ds}{dt} = \frac{g}{2} \cdot 2t = gt \]

### Step 4: Write the final expression for velocity
Thus, the velocity \( v \) at any time \( t \) is given by:

\[ v = gt \]

### Conclusion
The velocity of the body at any time \( t \) is:

\[ v = gt \]

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