Write an expression for the distance S covered in time t by a body which is initially at rest and starts moving with a constant acceleration a.

Explanation

To derive the expression for the distance \( S \) covered by a body that starts from rest and moves with a constant acceleration \( a \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Initial Conditions**:
- The body starts from rest, which means the initial velocity \( u = 0 \).
- The constant acceleration is given as \( a \).
- The time duration for which the body is moving is \( t \).

2. **Use the Equation of Motion**:
- The equation of motion that relates distance \( S \), initial velocity \( u \), time \( t \), and acceleration \( a \) is:
\[ S = ut + \frac{1}{2} a t^2 \]

3. **Substitute the Initial Velocity**:
- Since the body starts from rest, we substitute \( u = 0 \) into the equation:
\[ S = 0 \cdot t + \frac{1}{2} a t^2 \]

4. **Simplify the Expression**:
- The term \( 0 \cdot t \) equals zero, so we can simplify the equation:
\[ S = \frac{1}{2} a t^2 \]

5. **Final Expression**:
- Therefore, the expression for the distance \( S \) covered in time \( t \) by a body starting from rest with constant acceleration \( a \) is:
\[ S = \frac{1}{2} a t^2 \]