A stone is thrown vertically upwards with an initial speed `u` from the top of a tower, reaches the ground with a speed `3 u`. The height of the tower is :

Options

A. `3u^(2)//g`
B. `4u^(2)//g`
C. `6u^(2)//g`
D. `9u^(2)//g`

Explanation

To find the height of the tower from which a stone is thrown vertically upwards with an initial speed \( u \) and reaches the ground with a speed \( 3u \), we can use the equations of motion. Here’s a step-by-step solution:

### Step 1: Understand the problem
We have a stone thrown upwards with an initial speed \( u \) from the top of a tower of height \( h \). It reaches the ground with a final speed of \( 3u \).

### Step 2: Set up the equation of motion
We will use the following equation of motion:
\[ v^2 = u^2 + 2as \]
where:
- \( v \) is the final velocity,
- \( u \) is the initial velocity,
- \( a \) is the acceleration (which is \( g \), the acceleration due to gravity, and is positive in the downward direction),
- \( s \) is the displacement (which is equal to the height of the tower \( h \), and is also positive in the downward direction).

### Step 3: Assign values to variables
From the problem:
- Final velocity \( v = 3u \) (downward, so positive),
- Initial velocity \( u = -u \) (upward, so negative),
- Acceleration \( a = g \) (downward),
- Displacement \( s = h \) (downward).

### Step 4: Substitute values into the equation
Substituting the values into the equation:
\[ (3u)^2 = (-u)^2 + 2gh \]

### Step 5: Simplify the equation
Calculating the squares:
\[ 9u^2 = u^2 + 2gh \]
Now, rearranging the equation:
\[ 9u^2 - u^2 = 2gh \]
\[ 8u^2 = 2gh \]

### Step 6: Solve for height \( h \)
Dividing both sides by \( 2g \):
\[ h = \frac{8u^2}{2g} = \frac{4u^2}{g} \]

### Conclusion
The height of the tower is:
\[ h = \frac{4u^2}{g} \]