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`]

Options

A. `34 km`
B. `544 km`
C. `136 km`
D. `68 km`

Explanation

To solve the problem, we need to find the height \( h \) from which the piece of ice falls, given that it melts completely upon falling and that only one-quarter of the heat produced is absorbed by the ice.

### Step-by-Step Solution:

1. **Identify the energy conversion**:
When the ice falls from a height \( h \), its gravitational potential energy (PE) is converted into heat energy (Q) when it reaches the ground. The potential energy of the ice at height \( h \) is given by:
\[ PE = mgh \]
where \( m \) is the mass of the ice, \( g \) is the acceleration due to gravity, and \( h \) is the height.

2. **Heat absorbed by the ice**:
According to the problem, only one-quarter of the heat produced during the fall is absorbed by the ice. Therefore, the heat absorbed by the ice is:
\[ Q = \frac{1}{4} \times PE = \frac{1}{4} \times mgh \]

3. **Heat required to melt the ice**:
The heat required to completely melt the ice is given by the formula:
\[ Q_{melt} = mL \]
where \( L \) is the latent heat of fusion of ice. Given \( L = 3.4 \times 10^5 \, \text{J/kg} \), we can write:
\[ Q_{melt} = m \times 3.4 \times 10^5 \]

4. **Setting the equations equal**:
Since the heat absorbed by the ice is equal to the heat required to melt it, we can set the two equations equal to each other:
\[ \frac{1}{4} mgh = m \times 3.4 \times 10^5 \]

5. **Canceling mass \( m \)**:
We can cancel \( m \) from both sides of the equation (assuming \( m \neq 0 \)):
\[ \frac{1}{4} gh = 3.4 \times 10^5 \]

6. **Solving for height \( h \)**:
Rearranging the equation to solve for \( h \):
\[ h = \frac{4 \times 3.4 \times 10^5}{g} \]
Substituting \( g = 10 \, \text{N/kg} \):
\[ h = \frac{4 \times 3.4 \times 10^5}{10} = 1.36 \times 10^5 \, \text{m} \]

7. **Final conversion to kilometers**:
To convert meters to kilometers:
\[ h = 1.36 \times 10^5 \, \text{m} = 136 \, \text{km} \]

### Final Answer:
The value of \( h \) is \( 136 \, \text{km} \).