If a stone is relased from a balloon moving upwards with certain velocity `v_(0)` at certain height above earth's surface then velocity time curve of stone's motion can be best represented by : (g = constant)

Options

A.
B.
C.
D.

Explanation

To solve the problem, we need to analyze the motion of the stone after it is released from the balloon. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Initial Conditions
When the stone is released from the balloon, it has an initial upward velocity `v_0` (the same velocity as the balloon).

**Hint:** Remember that the stone will initially continue moving in the direction it was moving at the moment of release.

### Step 2: Identify the Forces Acting on the Stone
Once the stone is released, the only force acting on it is gravity, which pulls it downward with an acceleration `g`.

**Hint:** The stone is no longer influenced by the balloon's motion after release; it is now in free fall.

### Step 3: Analyze the Motion of the Stone
The stone will first move upward (since it has an initial upward velocity `v_0`), but as it ascends, the gravitational force will decelerate it until it reaches a maximum height where its velocity becomes zero.

**Hint:** The stone's upward motion will continue until the gravitational force counteracts its initial velocity.

### Step 4: Determine the Velocity-Time Relationship
1. **Upward Motion:** The stone will move upward until it reaches its maximum height. During this phase, its velocity decreases from `v_0` to `0`.
2. **Downward Motion:** After reaching the maximum height, the stone will start falling back down, gaining speed in the downward direction (negative velocity).

**Hint:** The velocity-time graph will show a decrease in velocity to zero and then a negative velocity as the stone falls back down.

### Step 5: Sketch the Velocity-Time Curve
The velocity-time curve will start at `v_0`, decrease linearly to `0` (at the maximum height), and then continue to decrease into negative values as the stone falls back down.

**Hint:** Since the acceleration due to gravity is constant, the velocity-time graph will be a straight line with a negative slope after the stone reaches its maximum height.

### Conclusion
Based on the analysis, the velocity-time curve of the stone's motion can be best represented by option 2, which shows a positive initial velocity that decreases to zero and then becomes negative.

**Final Answer:** The correct representation of the velocity-time curve is option 2.