A stone is dropped from the top of a tower. When it crosses a point 5 m below the top, another stone is let fall from a point 25 m below the top. Both stones reach the bottom of the tower simultaneously. Find the height of the tower. Take `g = 10 m//s^2.`

Explanation

To solve the problem, we need to find the height of the tower from which two stones are dropped. Let's break down the solution step by step.

### Step 1: Define the Variables
Let the height of the tower be \( H \) meters.
- The first stone (Stone A) is dropped from the top of the tower and crosses a point 5 m below the top.
- The second stone (Stone B) is dropped from a point 25 m below the top of the tower.

### Step 2: Analyze the Motion of Stone A
When Stone A is 5 m below the top, it has fallen a distance of 5 m. We can use the equation of motion to find its velocity at this point:

\[ v^2 = u^2 + 2as \]
Where:
- \( u = 0 \) (initial velocity, since it was dropped)
- \( a = g = 10 \, \text{m/s}^2 \) (acceleration due to gravity)
- \( s = 5 \, \text{m} \) (distance fallen)

Substituting the values:

\[ v^2 = 0 + 2 \cdot 10 \cdot 5 \]
\[ v^2 = 100 \implies v = 10 \, \text{m/s} \]

### Step 3: Time Taken by Stone A to Reach the Bottom
Now, Stone A will fall from 5 m below the top to the bottom of the tower. The distance it has to fall is \( H - 5 \) m. We can use the equation of motion again:

\[ s = ut + \frac{1}{2} a t^2 \]
Where:
- \( s = H - 5 \)
- \( u = 10 \, \text{m/s} \) (velocity at 5 m below the top)
- \( a = 10 \, \text{m/s}^2 \)

Substituting the values:

\[ H - 5 = 10t + \frac{1}{2} \cdot 10 \cdot t^2 \]
\[ H - 5 = 10t + 5t^2 \quad \text{(Equation 1)} \]

### Step 4: Analyze the Motion of Stone B
Stone B is dropped from a point 25 m below the top, which means it falls a distance of \( H - 25 \) m. The initial velocity of Stone B is 0.

Using the equation of motion:

\[ s = ut + \frac{1}{2} a t^2 \]
Where:
- \( s = H - 25 \)
- \( u = 0 \)
- \( a = 10 \, \text{m/s}^2 \)

Substituting the values:

\[ H - 25 = 0 + \frac{1}{2} \cdot 10 \cdot t^2 \]
\[ H - 25 = 5t^2 \quad \text{(Equation 2)} \]

### Step 5: Solve the Equations
Now we have two equations:
1. \( H - 5 = 10t + 5t^2 \)
2. \( H - 25 = 5t^2 \)

From Equation 2, we can express \( H \):

\[ H = 5t^2 + 25 \]

Substituting this expression for \( H \) into Equation 1:

\[ (5t^2 + 25) - 5 = 10t + 5t^2 \]
\[ 5t^2 + 20 = 10t + 5t^2 \]

Now, simplifying:

\[ 20 = 10t \]
\[ t = 2 \, \text{s} \]

### Step 6: Find the Height of the Tower
Now, substitute \( t = 2 \) back into Equation 2 to find \( H \):

\[ H - 25 = 5(2^2) \]
\[ H - 25 = 20 \]
\[ H = 45 \, \text{m} \]

### Final Answer
The height of the tower is **45 meters**.