Question

An ice block at $ 0^\circ C $ is dropped from the height $ h $ above the ground. What should be the value of $ h $ so that it just melts completely by the time it reaches the bottom assuming the loss of whole gravitational potential energy is used as heat by the ice? [Given: $ {L_f}\; = {\text{ }}80{\text{ }}cal/gm $ ]
A) $ 33.6{\text{ }}m $
B) $ 33.6{\text{ }}km $
C) 8 m
D) 8 km

Answer 1

Hint: When the ice block drops towards the ground, there will be a decrease in the potential energy of the ice block. This energy will be transferred to melting the ice block.

Formula used: In this solution, we will use the following formulae:
-Potential energy of an object: $ U = mgh $ where $ m $ is the mass of the block, $ g $ is the gravitational acceleration, and $ h $ is the height of the object.
-Latent heat energy of an object: $ Q = mL $ where $ m $ is the mass of the object and $ L $ is the latent heat constant.

Complete step by step answer:
We’ve been given that an ice block at $ 0^\circ C $ is dropped from a height $ h $ above the ground and as it falls, it melts such that the energy for its melting is provided by the decrease in potential energy of the object.
We know that the potential energy of the block is given as $ U = mgh $ and the heat required for melting the ice block is given as $ Q = mL $ .
assuming the loss of whole gravitational potential energy is used as heat by the ice, we can write
 $ mgh = mL $
Which gives us the height of the object as
 $ h = \dfrac{{{L_f}}}{g} $
Substituting the value of $ {L_f}\; = {\text{ }}80{\text{ }}cal/gm = 80 \times 4.2 \times 1000\,J/kg $ and $ g = 10\,m/{s^2} $ , we get
 $ h = \dfrac{{80 \times 4.2 \times 1000}}{{10}} = 33.6\,{\text{km}} $
Hence the height required for the ice block to melt will be $ h = 33.6\,km $ by the time it will reach the ground.
So the correct choice is option (B).

Note:
Here we have neglected any other forms of energy losses due to air resistance etc. When using the latent heat of melting ice, we must first convert it into its SI units otherwise we will get the answer as an option (C) which is incorrect.