A piece of ice falls from a height `h` so that it melts completely. Only one-quarter of the heat produced is absobed by the ice and all energy of ice gets converted into heat during its fall. The value of `h` is
[Latent heat of ice is `3.4 xx 10^(5) J//kg` and `g = 10 N//kg`]

Answer

To solve the problem, we need to find the height \( h \) from which a piece of ice falls and melts completely when it hits the ground. We know that only one-quarter of the gravitational potential energy is used to melt the ice, and all the energy from the fall is converted into heat.

### Step-by-Step Solution:

1. **Identify the Energy Conversion**:
- The gravitational potential energy (GPE) of the ice when it falls from height \( h \) is given by:
\[ \text{GPE} = mgh \]
- Here, \( m \) is the mass of the ice, \( g \) is the acceleration due to gravity, and \( h \) is the height.

2. **Determine the Heat Absorbed by Ice**:
- According to the problem, only one-quarter of the gravitational potential energy is absorbed by the ice for melting:
\[ \text{Heat absorbed} = \frac{mgh}{4} \]

3. **Relate Heat Absorbed to Latent Heat**:
- The heat required to melt the ice is given by the formula:
\[ \text{Heat} = mL \]
- Where \( L \) is the latent heat of fusion of ice. From the problem, we know \( L = 3.4 \times 10^5 \, \text{J/kg} \).

4. **Set Up the Equation**:
- Since the heat absorbed by the ice to melt it is equal to the heat required for melting, we can set the two expressions equal:
\[ \frac{mgh}{4} = mL \]
- We can cancel \( m \) from both sides (assuming \( m \neq 0 \)):
\[ \frac{gh}{4} = L \]

5. **Solve for Height \( h \)**:
- Rearranging the equation gives us:
\[ h = \frac{4L}{g} \]
- Now, substitute the values of \( L \) and \( g \):
\[ h = \frac{4 \times (3.4 \times 10^5)}{10} \]

6. **Calculate \( h \)**:
- Performing the calculation:
\[ h = \frac{13.6 \times 10^5}{10} = 1.36 \times 10^5 \, \text{m} = 136 \, \text{km} \]

### Final Answer:
The height \( h \) from which the ice falls is \( 136 \, \text{km} \).