A stone is dropped from the top of tower. When it has fallen by `5 m` from the top, another stone is dropped from a point `25m` below the top. If both stones reach the ground at the moment, then height of the tower from grounds is : (take `g=10 m//s^(2)`)

Options

A. `45 m`
B. `50 m`
C. `60 m`
D. `65 m`

Explanation

To solve the problem step by step, we will analyze the motion of both stones dropped from the tower.

### Step 1: Understand the problem
We have a tower of height \( h \). A stone is dropped from the top of the tower. After it has fallen \( 5 \, m \), another stone is dropped from a point \( 25 \, m \) below the top of the tower. Both stones reach the ground at the same time.

### Step 2: Define the variables
Let:
- \( h \) = height of the tower
- \( g = 10 \, m/s^2 \) (acceleration due to gravity)
- \( t_1 \) = time taken by the first stone to reach the ground
- \( t_2 \) = time taken by the second stone to reach the ground

### Step 3: Analyze the first stone
The first stone is dropped from the top of the tower. When it has fallen \( 5 \, m \), the distance it has left to fall is \( h - 5 \, m \).

Using the equation of motion:
\[ s = ut + \frac{1}{2} g t^2 \]
where \( u = 0 \) (initial velocity), we can write:
\[ h - 5 = \frac{1}{2} g t_1^2 \]
Substituting \( g = 10 \):
\[ h - 5 = \frac{1}{2} \times 10 \times t_1^2 \]
\[ h - 5 = 5 t_1^2 \quad \text{(1)} \]

### Step 4: Analyze the second stone
The second stone is dropped from \( 25 \, m \) below the top, which means it starts falling from \( h - 25 \, m \). The time taken for this stone to reach the ground is \( t_2 \).

Using the same equation of motion:
\[ h - 25 = \frac{1}{2} g t_2^2 \]
Substituting \( g = 10 \):
\[ h - 25 = 5 t_2^2 \quad \text{(2)} \]

### Step 5: Relate the times
Since the first stone falls for \( t_1 \) seconds and the second stone falls for \( t_2 \) seconds, and the second stone is dropped \( 1 \, s \) after the first stone, we have:
\[ t_2 = t_1 - 1 \quad \text{(3)} \]

### Step 6: Substitute and solve
Substituting equation (3) into equation (2):
\[ h - 25 = 5(t_1 - 1)^2 \]
Expanding this:
\[ h - 25 = 5(t_1^2 - 2t_1 + 1) \]
\[ h - 25 = 5t_1^2 - 10t_1 + 5 \]
Rearranging gives:
\[ h = 5t_1^2 - 10t_1 + 30 \quad \text{(4)} \]

### Step 7: Equate equations (1) and (4)
From equation (1):
\[ h = 5t_1^2 + 5 \]
Setting equations (4) and (1) equal:
\[ 5t_1^2 + 5 = 5t_1^2 - 10t_1 + 30 \]
Cancelling \( 5t_1^2 \) from both sides:
\[ 5 = -10t_1 + 30 \]
Solving for \( t_1 \):
\[ 10t_1 = 30 - 5 \]
\[ 10t_1 = 25 \]
\[ t_1 = 2.5 \, s \]

### Step 8: Find the height of the tower
Substituting \( t_1 = 2.5 \) back into equation (1):
\[ h - 5 = 5(2.5)^2 \]
\[ h - 5 = 5 \times 6.25 \]
\[ h - 5 = 31.25 \]
\[ h = 31.25 + 5 = 36.25 \, m \]

### Step 9: Conclusion
Thus, the height of the tower is approximately \( 36.25 \, m \).