A stone projected vertically up with velocity v from the top of a tower reaches the ground with velocity 2v. The height of the tower is

Options

A. `(v^(2))/(2g)`
B. `(3v^(2))/(2g)`
C. `(3v^(2))/(g)`
D. `(v^(2))/(g)`

Answer

To solve the problem, we need to find the height of the tower from which a stone is projected vertically upwards with an initial velocity \( v \) and reaches the ground with a final velocity of \( 2v \).

### Step-by-Step Solution:

1. **Understanding the Motion**:
- The stone is projected upwards from the top of the tower with an initial velocity \( v \).
- It reaches a maximum height \( h \) before falling back down and ultimately hitting the ground with a final velocity of \( 2v \).

2. **Finding the Maximum Height \( h \)**:
- We can use the first equation of motion for the upward journey to find the height \( h \):
\[ v^2 = u^2 + 2as \]
Here, \( v = 0 \) (the final velocity at the maximum height), \( u = v \) (initial velocity), \( a = -g \) (acceleration due to gravity), and \( s = h \) (the height reached).
- Substituting the values:
\[ 0 = v^2 - 2gh \]
Rearranging gives:
\[ 2gh = v^2 \quad \Rightarrow \quad h = \frac{v^2}{2g} \]

3. **Finding the Total Height from the Tower to the Ground**:
- Now, we need to consider the motion from the maximum height \( h \) back to the ground. The total height from the top of the tower to the ground is \( H + h \), where \( H \) is the height of the tower.
- Using the second equation of motion for the downward journey:
\[ v_f^2 = v_i^2 + 2as \]
Here, \( v_f = 2v \) (final velocity), \( v_i = 0 \) (initial velocity at the maximum height), \( a = g \) (acceleration), and \( s = H + h \) (total distance fallen).
- Substituting the values:
\[ (2v)^2 = 0 + 2g(H + h) \]
This simplifies to:
\[ 4v^2 = 2g(H + h) \]
Dividing both sides by 2:
\[ 2v^2 = g(H + h) \]

4. **Substituting for \( h \)**:
- We already found \( h = \frac{v^2}{2g} \). Substitute this into the equation:
\[ 2v^2 = g\left(H + \frac{v^2}{2g}\right) \]
- Simplifying gives:
\[ 2v^2 = gH + \frac{v^2}{2} \]
- Rearranging yields:
\[ 2v^2 - \frac{v^2}{2} = gH \]
\[ \frac{4v^2}{2} - \frac{v^2}{2} = gH \]
\[ \frac{3v^2}{2} = gH \]

5. **Finding the Height of the Tower \( H \)**:
- Finally, solving for \( H \):
\[ H = \frac{3v^2}{2g} \]

### Conclusion:
The height of the tower is \( H = \frac{3v^2}{2g} \).