A stone projected at angle `theta` with horizontal from the roof of a tall building falls on the ground after three seconds. Two second after the projection it was again at the level of projection. Then the height of the building

Options

A. `5m`
B. 25 m
C. 20m
D. 15m

Explanation

To solve the problem step by step, we need to analyze the motion of the stone projected from the roof of a tall building.

### Step 1: Understand the Problem
We know that a stone is projected at an angle θ with the horizontal from the roof of a tall building. It falls to the ground after 3 seconds. It reaches the level of projection again after 2 seconds. We need to find the height of the building (h).

### Step 2: Analyze the Motion
The stone is projected with an initial velocity \( u \). We can break this velocity into two components:
- Horizontal component: \( u \cos \theta \)
- Vertical component: \( u \sin \theta \)

### Step 3: Time of Flight
The total time of flight until the stone hits the ground is given as 3 seconds. The stone returns to the level of projection after 2 seconds. Therefore, the time taken to fall from the level of projection to the ground is:
\[ t_{\text{fall}} = 3 \text{ seconds} - 2 \text{ seconds} = 1 \text{ second} \]

### Step 4: Vertical Motion Equations
Using the vertical motion equation, we can express the height of the building (h) in terms of the vertical motion:
\[ h = u \sin \theta \cdot t_{\text{total}} - \frac{1}{2} g t_{\text{fall}}^2 \]
Where:
- \( t_{\text{total}} = 3 \) seconds (total time of flight)
- \( t_{\text{fall}} = 1 \) second (time taken to fall to the ground)
- \( g = 10 \, \text{m/s}^2 \) (acceleration due to gravity)

### Step 5: Calculate the Vertical Component
From the information given, we know that the stone reaches the level of projection after 2 seconds. The time taken to reach this level can be expressed as:
\[ t_{\text{up}} = \frac{2u \sin \theta}{g} \]
Setting this equal to 2 seconds gives:
\[ 2 = \frac{2u \sin \theta}{g} \implies u \sin \theta = 10 \, \text{m/s} \]

### Step 6: Substitute Back to Find Height
Now substituting \( u \sin \theta = 10 \) into the height equation:
\[ h = 10 \cdot 3 - \frac{1}{2} \cdot 10 \cdot (1^2) \]
Calculating this gives:
\[ h = 30 - 5 = 25 \, \text{meters} \]

### Conclusion
The height of the building is \( h = 25 \, \text{meters} \).