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

Explanation

To solve the problem, we will use the principle of conservation of energy. The energy gained by the ice during its fall will be equal to the energy required to melt the ice.

### Step-by-Step Solution:

1. **Identify the Energy Gained by the Ice**:
When the ice falls from a height \( h \), it gains gravitational potential energy given by the formula:
\[ E_{\text{gained}} = mgh \]
where:
- \( m \) = mass of the ice (in kg)
- \( g \) = acceleration due to gravity (10 N/kg)
- \( h \) = height from which the ice falls (in meters)

2. **Determine the Energy Absorbed by the Ice**:
According to the problem, only one-quarter of the energy gained by the ice is used to melt it. Therefore, the energy absorbed by the ice is:
\[ E_{\text{absorbed}} = \frac{1}{4} mgh \]

3. **Relate Absorbed Energy to Latent Heat**:
The energy required to melt the ice is given by the formula:
\[ E_{\text{melt}} = mL_f \]
where:
- \( L_f \) = latent heat of fusion of ice (given as \( 3.4 \times 10^5 \, \text{J/kg} \))

4. **Set the Two Energies Equal**:
Since the energy absorbed by the ice is used to melt it, we can set the two expressions equal:
\[ \frac{1}{4} mgh = mL_f \]

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

6. **Solve for Height \( h \)**:
Rearranging the equation to solve for \( h \):
\[ h = \frac{4L_f}{g} \]

7. **Substitute the Known Values**:
Now, substitute the values of \( L_f \) and \( g \):
\[ h = \frac{4 \times 3.4 \times 10^5}{10} \]

8. **Calculate \( h \)**:
\[ h = \frac{13.6 \times 10^5}{10} = 1.36 \times 10^5 \, \text{m} = 136000 \, \text{m} = 136 \, \text{km} \]

### Final Answer:
The value of \( h \) is approximately **136 km**.