Question

A block of ice with mass m falls into a lake. After impact, a mass of ice \[\dfrac{m}{5}\] melts. Both the block of ice and the lake have a temperature of \[0^\circ C\]. If L represents the heat of fusion, the minimum distance the ice fell before striking the surface is
A. \[\dfrac{L}{{5g}}\]
B. \[\dfrac{{5L}}{g}\]
C. \[\dfrac{{gL}}{{5m}}\]
D. \[\dfrac{{mL}}{{5g}}\]

Answer 1

Hint: The melting of the ice block is due to the latent heat of fusion of the ice at constant temperature. Use the law of conservation of energy since the loss in the potential energy of the ice block is equal to the gain in the kinetic energy. The same kinetic energy is used to melt the block of ice.

Formula used:
Latent heat of fusion, \[Q = mL\]
Here, L is the latent heat of fusion of the substance.
Potential energy, \[U = mgh\],
where, m is the mass, g is the acceleration due to gravity and h is the height.

Complete step by step answer:
We have given that the temperature of both the lake and ice block is the same. Therefore, there will not be heat energy transfer between the lake to the ice block or vice versa. The melting of the ice block is due to the latent heat of fusion of the ice at constant temperature.Let h be the height from which the ice block is fell.

As the ice falls, its potential energy is converted into the kinetic energy and the kinetic energy obtained by the block is used to melt the ice of mass \[\dfrac{m}{5}\] at the same temperature. The latent heat energy required to melt the substance of mass m is expressed as,
\[Q = mL\]
Here, L is the latent heat of fusion of the substance.
Therefore, we can write,
\[mgh = \dfrac{m}{5}L\]
\[ \Rightarrow gh = \dfrac{L}{5}\]
\[ \therefore h = \dfrac{L}{{5g}}\]

So, the correct answer is option A.

Note: Instead of using the term latent heat of fusion, one should call it latent heat of melting of ice. The latent heat of melting and latent heat of fusion of ice requires the same heat energy. If the block of ice strikes the lake of ice, the temperature of the ice block would have increased due to impact.