Question

A piece of ice falls from a height $h$ so that it melts completely. Only one-quarter of the heat produced is absorbed 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 $3.4\times {{10}^{5}}J/Kg$ and $g=10N/kg$]
A. 34 km
B. 544 km
C. 136 km
D. 68 km

Answer 1

We have an ice falling from a height ‘h’. We are asked to find the value of ‘h’ using the given parameters. We know that when an object falls from a height its gravitational potential energy is converted into heat. We equate this statement and by substituting the given values we get the height of the fall.

Formula used: $mgh=mL$

Complete step by step answer:

In the question it is said that an ice piece falls down from a height ‘h’ and it is melted completely.
During the fall, only ${{\dfrac{1}{4}}^{th}}$ of the heat is absorbed by the ice and the rest of the energy is converted into heat.
We are given the latent heat of ice and the value of g.
Latent heat of ice$=3.4\times {{10}^{5}}J/k$
$g=10N/kg$
Now, let us consider ‘m’ to be the mass of ice.
When ice falls down from the height ‘h’, we know that its potential energy is converted into heat energy.
I.e. gravitational potential energy is equivalent to the latent heat of ice.
Since the mass of ice is ‘m’ here,
Latent heat can be written as
$'mL'$ , Where ‘L’ is the latent heat of ice
Therefore we can say that,
$mgh=mL$ , Where $'mgh'$ is the potential energy and $'mL'$ is the latent heat.
It is said that only one-fourth of the heat is absorbed by the ice during fall.
Therefore we can say that
$\dfrac{mgh}{4}=mL$
By solving the above equation, we get ‘h’ as,
$h=\dfrac{4L}{g}$
Substituting the values of ‘L’ and ‘g’, we get
$\begin{aligned} & h=\dfrac{4\times 3.4\times {{10}^{5}}J/kg}{10N/kg} \\\ & h=136\times {{10}^{3}}m \\\ & h=136km \\\ \end{aligned}$
Therefore, the ice falls from a height h=136 km.

So, the correct answer is “Option C”.

Note: Latent heat of a substance is the amount of heat that is required to convert a solid into liquid or vapor, or a liquid into vapor state without changing the temperature. It is also stated as the energy released or absorbed by a body or a system during an isothermal process.
When a body falls from a height its potential energy gets converted into kinetic energy. This energy will be absorbed by the body as heat.
The energy stored in an object due to its position relative to other objects is that body’s potential energy.