Question

A waterfall is \[100\,m\] high. How much warmer will the water be after the fall? Assume that all the work gets converted into heat energy. Take the average value of $g$ to be \[980\,cm{s^{ - 2}}\].

Answer 1

Hint: Use the law of conversion of energy to solve the problem. This law states that the energy of a system is always conserved or constant under the act of any conservative force. The heat absorbed by a body is given by the product of its mass, specific heat and the temperature difference between the states.

Formula used:
The potential energy of a body is given by the formula,
\[P.E = mgh\]
where, \[P.E\] is the potential energy of the body \[m\] is the total mass of the body and \[h\] is the height of the body.
The heat absorbed by a body is given by,
\[Q = mst\]
where, \[s\] is the specific heat of the object and t is temperature difference.

Complete step by step answer:
We know that energy is conserved for any conservative system. Now, if water falls from a height of \[100\,m\] the potential energy will be converted into heat energy. Now, let the mass be $m$, the specific heat is s and temperature different is m. Now, if the water falls from a height of 100 m the potential energy will be,
\[P.E = 100\,mg\].
Now this energy will be absorbed by the water as heat energy.

So, this will be equal to,
\[mg \times 100 = m \times 4.2 \times t\]..............[ where, \[4.2\,Jg{m^{ - 1}}\] or \[4.2 \times 1000\,Jk{g^{ - 1}}\] is specific heat capacity of water]
\[t = \dfrac{{100 \times 9.8}}{{4.2 \times 1000}}\]...........[ where, \[g = 980\,cm{s^{ - 2}} = 9.8\,m{s^{ - 2}}\]]
\[\therefore t = 0.233\]
So, the temperature difference between the top and the groundwater will be \[0.233K\]or \[{0.233^ \circ }C\].

Hence, the water will get \[{0.233^ \circ }C\] or \[0.233\,K\] warmer from falling from a height of \[100\,m\].

Note: While calculating the temperature difference take the units of every quantity with care, failing to do so will get incorrect results of temperature difference. Either use all the units SI system or use all the units in CGS system to calculate. The temperature difference in Kelvin scale and Celsius scale are the same.