Question

Hailstones fall from a certain height. If only 1 percent of the hailstones melt on reaching the ground, find the height from which they fall.
$(g=10m{{s}^{-2}},L=80cal{{g}^{-1}},J=4.2J/cal)$
$\begin{align}
  & (A)336m \\
 & (B)236m \\
 & (C)436m \\
 & (D)536m \\
\end{align}$

Answer 1

Hint: Hailstones fall from a certain height, so there must be a decrease in their potential energy after they have reached the ground. This energy is in turn absorbed by the 1 percent of hailstones that melt on reaching the ground. We shall use this fact as a basis to solve the given problem.

Complete answer:
Let the height from which the hailstones fall be given by $h$meters. Also, let the mass of the hailstone when they are at a height be denoted by $m$.
Then, the mass of the hailstone that melt will be equal to:
$=$ 1 percent of $m$
$\begin{align}
  & =\dfrac{1}{100}\times m \\
 & =\dfrac{m}{100} \\
\end{align}$
Now, we shall find the decrease in potential energy of the hails after they have fallen. Since this decrease is only due to the gravitational force, its value will be equal to:
$=mgh$ [Let this expression be equation number (1)]
Also, the amount of heat required to melt the 1 percent of hailstones will be equal to:
$=\dfrac{m}{100}\times J\times {{L}_{ice}}$ [Let this expression be equation number (2)]
Where,
$J$ is the Joule’s constant whose value is given to us as:
$\Rightarrow J=4.2J/cal$
 $L$ is the Latent of ice whose value is given to us as:
$\Rightarrow L=80ca{{\lg }^{-1}}$
But we have to convert this into standard unit, that is equal to:
$\Rightarrow L=80\times {{10}^{3}}calk{{g}^{-1}}$
Now, since the decrease in potential energy of hailstones is the energy absorbed by a fraction of them to melt. Therefore, equation number (1) and (2) must be equal:
Thus, we can write:
$\Rightarrow mgh=\dfrac{m}{100}\times J\times L$
Putting the values of all the known terms and solving for the height $h$, we get:
$\begin{align}
  & \Rightarrow h=\dfrac{JL}{100g} \\
 & \Rightarrow h=\dfrac{4.2\times 80\times {{10}^{3}}}{100\times 10} \\
\end{align}$
On further simplifying, we get the required height as:
$\Rightarrow h=336m$
Hence, the height from which the hailstones are falling is 336m.

Hence, option (A) is the correct option.

Note:
When we started our problem, we assumed the mass of the hailstones. Now, this mass used in both the formulas but it eventually was cut out from the final equation. This proves, we need not worry if the value of any variable is not given in the problem. We should just stick to our concept and make sure we do not concur any calculational mistakes.