From what heigh should a piece of ice fall so that it melts completely ? Only one quarter of the heat produced is absorbed by the ice. Latent heat of ice is `3.4xx10^(5) j kg^(-1)` and `g=10 ms^(2)`.

Explanation

To solve the problem of determining the height from which a piece of ice should fall to melt completely, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We need to find the height from which ice must fall so that it completely melts upon impact. We know that only one quarter of the heat produced during the fall is absorbed by the ice.

2. **Identify Given Values**:
- Latent heat of ice, \( L = 3.4 \times 10^5 \, \text{J/kg} \)
- Acceleration due to gravity, \( g = 10 \, \text{m/s}^2 \)

3. **Calculate the Potential Energy**: The potential energy (PE) of the ice at height \( h \) is given by:
\[ PE = mgh \]
where \( m \) is the mass of the ice.

4. **Determine the Heat Absorbed by Ice**: Since only one quarter of the potential energy is absorbed by the ice, the heat absorbed by the ice is:
\[ Q = \frac{1}{4} mgh \]

5. **Heat Required to Melt the Ice**: The heat required to melt the ice is given by:
\[ Q_{\text{melt}} = mL \]

6. **Set the Heat Absorbed Equal to Heat Required**: For the ice to melt completely, the heat absorbed must equal the heat required to melt it:
\[ \frac{1}{4} mgh = mL \]

7. **Cancel Mass \( m \)**: Since mass \( m \) appears on both sides of the equation, we can cancel it out (assuming \( m \neq 0 \)):
\[ \frac{1}{4} gh = L \]

8. **Solve for Height \( h \)**: Rearranging the equation gives:
\[ h = \frac{4L}{g} \]

9. **Substitute Known Values**:
\[ h = \frac{4 \times 3.4 \times 10^5}{10} \]

10. **Calculate \( h \)**:
\[ h = \frac{13.6 \times 10^5}{10} = 1.36 \times 10^5 \, \text{m} \]

11. **Final Result**: Therefore, the height from which the ice should fall to melt completely is:
\[ h = 136000 \, \text{m} \text{ or } 136 \, \text{km} \]