A stone falling from the top of a vertical tower has descended `x` metre when another is dropped from a point `y` metre, below the top. If they fall from rest and reach the ground together, show that the height of the tower is `(x+y)^(2)/(4x) m`.

Explanation

To solve the problem, we need to analyze the motion of two stones falling from a tower. Let's denote the height of the tower as \( h \).

### Step-by-Step Solution:

1. **Understanding the Problem**:
- A stone is dropped from the top of a tower and falls a distance \( x \) meters.
- Another stone is dropped from a point \( y \) meters below the top of the tower.
- Both stones reach the ground at the same time.

2. **Determine the Time for the First Stone**:
- The first stone falls a distance \( x \) meters.
- Using the second equation of motion:
\[ s = ut + \frac{1}{2} a t^2 \]
where \( s = x \), \( u = 0 \) (initial velocity), and \( a = g \) (acceleration due to gravity).
- The equation simplifies to:
\[ x = \frac{1}{2} g t_1^2 \]
- Rearranging gives:
\[ t_1^2 = \frac{2x}{g} \quad \Rightarrow \quad t_1 = \sqrt{\frac{2x}{g}} \]

3. **Determine the Time for the Second Stone**:
- The second stone falls from a height of \( h - y \) meters.
- Using the same equation of motion:
\[ h - y = \frac{1}{2} g t_2^2 \]
- Rearranging gives:
\[ t_2^2 = \frac{2(h - y)}{g} \quad \Rightarrow \quad t_2 = \sqrt{\frac{2(h - y)}{g}} \]

4. **Relating the Times**:
- Since both stones reach the ground together, the time taken by the first stone to fall \( x \) meters plus the time taken to fall the remaining height \( h - x \) is equal to the time taken by the second stone:
\[ t = t_1 + t_2 \]
- Therefore:
\[ \sqrt{\frac{2h}{g}} = \sqrt{\frac{2x}{g}} + \sqrt{\frac{2(h - y)}{g}} \]

5. **Eliminating \( g \)**:
- Multiplying through by \( \sqrt{g} \) gives:
\[ \sqrt{2h} = \sqrt{2x} + \sqrt{2(h - y)} \]

6. **Squaring Both Sides**:
- Squaring both sides results in:
\[ 2h = 2x + 2(h - y) + 2\sqrt{2x} \sqrt{2(h - y)} \]
- Simplifying gives:
\[ 2h = 2x + 2h - 2y + 2\sqrt{2x} \sqrt{2(h - y)} \]
- Cancel \( 2h \) from both sides:
\[ 0 = 2x - 2y + 2\sqrt{2x} \sqrt{2(h - y)} \]

7. **Isolating \( h \)**:
- Rearranging gives:
\[ 2y = 2x + 2\sqrt{2x} \sqrt{2(h - y)} \]
- Dividing by 2:
\[ y = x + \sqrt{2x} \sqrt{2(h - y)} \]

8. **Final Rearrangement**:
- Isolate \( h \):
\[ h = \frac{(x + y)^2}{4x} \]

### Conclusion:
Thus, the height of the tower is given by:
\[ h = \frac{(x + y)^2}{4x} \, \text{meters} \]