Which of the following x-t graphs represents the distance-time variation of a body released from the top of a buiding?

Options

A.
B.
C.
D.

Answer

To determine which x-t graph represents the distance-time variation of a body released from the top of a building, we can analyze the motion of the body under the influence of gravity. Here’s a step-by-step solution:

### Step 1: Understand the Motion
When a body is released from the top of a building, it is in free fall. This means it experiences constant acceleration due to gravity (approximately \(9.81 \, \text{m/s}^2\) downward).

**Hint:** Remember that free fall implies constant acceleration.

### Step 2: Identify the Initial Conditions
Since the body is released (not thrown), its initial velocity (\(u\)) is 0 m/s. The only force acting on it is gravity.

**Hint:** The initial velocity is crucial for determining the shape of the graph.

### Step 3: Use the Kinematic Equation
The displacement (\(x\)) of the body can be described by the kinematic equation:
\[ x = ut + \frac{1}{2} a t^2 \]
Substituting \(u = 0\) and \(a = g\) (acceleration due to gravity):
\[ x = 0 \cdot t + \frac{1}{2} g t^2 = \frac{1}{2} g t^2 \]

**Hint:** Recognize that this equation is quadratic in form.

### Step 4: Analyze the Graph
The equation \(x = \frac{1}{2} g t^2\) indicates that the displacement \(x\) is proportional to the square of the time \(t\). The graph of a quadratic equation is a parabola that opens upwards.

**Hint:** Look for a parabolic shape in the options provided.

### Step 5: Eliminate Incorrect Options
- The first two options are straight lines, which represent uniform motion (constant velocity), so they cannot represent a body in free fall.
- The fourth option shows distance decreasing with time, which is not possible for a body in free fall.

**Hint:** Distance can only increase or remain constant; it cannot decrease in this scenario.

### Step 6: Identify the Correct Option
The only option left that shows a parabolic curve and represents increasing distance over time is the third option.

**Hint:** Confirm that the graph matches the expected behavior of the body in free fall.

### Conclusion
The correct x-t graph that represents the distance-time variation of a body released from the top of a building is **Option 3**.

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