A block of ice with mass m falls into a lake. After impact, a mass of ice `(m)/(5)` melts. Both the block of ice and the lake have a temperature of `0^(@)C`. If L represents the latent heat of fusion, the distance the ice falls before striking the surface is

Options

A. `(L)/(5g)`
B. `(5L)/(g)`
C. `(gL)/(5m)`
D. `(mL)/(5g)`

Answer

To find the distance the block of ice falls before striking the surface of the lake, we can use the principle of conservation of energy. The potential energy lost by the ice block as it falls is converted into the latent heat required to melt a portion of the ice.

### Step-by-Step Solution:

1. **Identify the potential energy lost**:
When the block of ice falls from a height \( h \), the potential energy lost is given by:
\[ PE = mgh \]
where \( m \) is the mass of the ice block, \( g \) is the acceleration due to gravity, and \( h \) is the height from which it falls.

2. **Identify the heat required to melt the ice**:
The mass of ice that melts after the impact is \( \frac{m}{5} \). The heat required to melt this mass of ice is given by:
\[ Q = \frac{m}{5} L \]
where \( L \) is the latent heat of fusion.

3. **Set the potential energy equal to the heat required**:
According to the conservation of energy, the potential energy lost by the ice block is equal to the heat required to melt the ice:
\[ mgh = \frac{m}{5} L \]

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

5. **Solve for height \( h \)**:
Rearranging the equation to solve for \( h \):
\[ h = \frac{L}{5g} \]

Thus, the distance the ice falls before striking the surface of the lake is:
\[ h = \frac{L}{5g} \]