A stone is thrown upwards from the top of a tower with some initial speed and it reaches the ground in `16` seconds. Now it is thrown with the same initial speed downward and it reaches the ground in `9 sec`. In how much time will it reach the ground if the stone is allowed to fall freely under gravity from the same place ?

Options

A. 25 s
B. 12.5 s
C. 7 s
D. 12 s

Explanation

To solve the problem, we need to analyze the motion of the stone in three different scenarios: when it is thrown upwards, thrown downwards, and dropped freely.

### Step-by-Step Solution:

1. **Define Variables**:
- Let \( u \) be the initial speed of the stone.
- Let \( H \) be the height of the tower.
- Let \( g \) be the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)).
- Let \( t_1 = 16 \, \text{s} \) (time taken when thrown upwards).
- Let \( t_2 = 9 \, \text{s} \) (time taken when thrown downwards).

2. **Equation for Upward Throw**:
When the stone is thrown upwards, the equation of motion is:
\[ H = u t_1 - \frac{1}{2} g t_1^2 \]
(Equation 1)

3. **Equation for Downward Throw**:
When the stone is thrown downwards, the equation of motion is:
\[ H = u t_2 + \frac{1}{2} g t_2^2 \]
(Equation 2)

4. **Set Up the System of Equations**:
From Equation 1:
\[ H = u \cdot 16 - \frac{1}{2} g \cdot 16^2 \]
From Equation 2:
\[ H = u \cdot 9 + \frac{1}{2} g \cdot 9^2 \]

5. **Equate the Two Expressions for H**:
Set the two expressions for \( H \) equal to each other:
\[ u \cdot 16 - \frac{1}{2} g \cdot 16^2 = u \cdot 9 + \frac{1}{2} g \cdot 9^2 \]

6. **Rearranging the Equation**:
Rearranging gives:
\[ u \cdot (16 - 9) = \frac{1}{2} g \cdot (16^2 - 9^2) \]
\[ 7u = \frac{1}{2} g (256 - 81) \]
\[ 7u = \frac{1}{2} g \cdot 175 \]
\[ u = \frac{g \cdot 175}{14} \]

7. **Substituting g**:
Substituting \( g = 9.81 \, \text{m/s}^2 \):
\[ u = \frac{9.81 \cdot 175}{14} \approx 122.5 \, \text{m/s} \]

8. **Finding H**:
Substitute \( u \) back into either Equation 1 or Equation 2 to find \( H \):
\[ H = u \cdot 16 - \frac{1}{2} g \cdot 16^2 \]
\[ H = 122.5 \cdot 16 - \frac{1}{2} \cdot 9.81 \cdot 256 \]

9. **Calculate H**:
\[ H = 1960 - 1254.72 \approx 705.28 \, \text{m} \]

10. **Free Fall Time Calculation**:
For free fall, the equation is:
\[ H = \frac{1}{2} g t^2 \]
Substituting \( H \):
\[ 705.28 = \frac{1}{2} \cdot 9.81 \cdot t^2 \]
\[ t^2 = \frac{705.28 \cdot 2}{9.81} \approx 143.5 \]
\[ t \approx \sqrt{143.5} \approx 12 \, \text{s} \]

### Final Answer:
The stone will take approximately **12 seconds** to reach the ground if it is allowed to fall freely under gravity from the same place.