The displacement of a body is given by `s=(1)/(2)g t^(2)` where g is acceleration due to gravity. The velocity of the body at any time t is

Options

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

Answer

To find the velocity of the body at any time \( t \) given the displacement \( s = \frac{1}{2} g t^2 \), we can follow these steps:

### Step 1: Understand the given displacement equation
The displacement of the body is given by:
\[ s = \frac{1}{2} g t^2 \]
where \( g \) is the acceleration due to gravity.

### Step 2: Identify initial conditions
From the equation, we can infer that the body starts from rest, which means the initial velocity \( u = 0 \).

### Step 3: Use the relationship between displacement, velocity, and time
The velocity \( v \) of the body at any time \( t \) can be derived from the displacement. We know that velocity is the derivative of displacement with respect to time:
\[ v = \frac{ds}{dt} \]

### Step 4: Differentiate the displacement equation
Now, we differentiate the displacement equation \( s = \frac{1}{2} g t^2 \) with respect to \( t \):
\[ v = \frac{d}{dt}\left(\frac{1}{2} g t^2\right) \]
Using the power rule of differentiation:
\[ v = \frac{1}{2} g \cdot 2t = g t \]

### Step 5: Write the final expression for velocity
Thus, the velocity of the body at any time \( t \) is given by:
\[ v = g t \]

### Summary
The velocity of the body at any time \( t \) is \( v = g t \).

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