A stone is allowed to fall freely from a certain Height. Neglecting air resistance, which graph represents the variation of velocity `v` with time `t` ?

Options

A.
B.
C.
D.

Explanation

To solve the problem of determining the graph that represents the variation of velocity \( v \) with time \( t \) for a stone falling freely from a height, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Motion**: The stone is falling freely under the influence of gravity. We neglect air resistance, which means the only force acting on the stone is gravity.

2. **Identify Initial Conditions**: When the stone is released from rest, its initial velocity \( u = 0 \).

3. **Use the Equation of Motion**: We can use the first equation of motion, which is given by:
\[ v = u + at \]
where:
- \( v \) is the final velocity,
- \( u \) is the initial velocity,
- \( a \) is the acceleration,
- \( t \) is the time.

4. **Substitute Known Values**: Since the initial velocity \( u = 0 \) and the acceleration \( a \) is equal to the acceleration due to gravity \( g \), we can rewrite the equation as:
\[ v = 0 + gt \]
or simply:
\[ v = gt \]

5. **Analyze the Equation**: The equation \( v = gt \) indicates that the velocity \( v \) increases linearly with time \( t \). The slope of the graph will be equal to \( g \), which is a positive constant.

6. **Graph Characteristics**:
- The graph starts from the origin (0,0) because the initial velocity is zero.
- As time increases, the velocity also increases, resulting in a straight line with a positive slope.

7. **Select the Correct Graph**: Based on the characteristics derived from the equation \( v = gt \), the correct graph will be a straight line starting from the origin and sloping upwards, indicating that velocity increases with time.

### Conclusion:
The graph that represents the variation of velocity \( v \) with time \( t \) for a stone falling freely from a height, neglecting air resistance, is a straight line passing through the origin with a positive slope.