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: The height of the Niagara Falls is 50 metres, $(1 c a l=4.2 J)$. Assume its mechanical energy can be completely converted into heat energy.
This question has multiple correct options
(A) Heat energy gained by each gram of water is $49 \times 10^{5}$ cal
(B) Rise in temperature of water is $0.166^{\circ} \mathrm{C} / \mathrm{g}$
(C) Rise in temperature of water is $0.12^{\circ} \mathrm{C}$
(D) Heal energy gained is 500 joule/g

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Last updated date: 30th May 2024
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Answer
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Hint: It is required to know that the energy acquired by the objects upon which work is done is known as mechanical energy. Mechanical energy is the energy that is possessed by an object due to its motion or due to its position. Mechanical energy can be either kinetic energy (energy of motion) or potential energy (stored energy of position). Mechanical energy is the sum of kinetic and potential energy in an object that is used to do work. In other words, it is energy in an object due to its motion or position, or both. Mechanical energy is a form of energy. It is all the energy that an object has because of its motion and its position. All living things and all machines use mechanical energy to do work.

Complete step by step answer
We know that:
Height of Niagara Falls $(\mathrm{h})=50 \mathrm{m}$
Let us consider that $\mathrm{m}$ is the mass of a waterfall.
Decrease in potential energy of waterfall is given as:
$\mathrm{U}=\mathrm{mgh}$
Hence after we put the values, we can write them as:
$\Rightarrow \mathrm{U}=0.018 \times 9.8 \times 50=$$8.82 \mathrm{J}$
(mass of 1 mole of water $=18 \mathrm{g}=0.018 \mathrm{kg}$ )
Now this potential energy is converted into heat. Let $\theta$ be the rise in temperature. Therefore,
$\mathrm{mgh}=\mathrm{ms} \theta$
On further evaluation we get:
$\Rightarrow \theta=\dfrac{\operatorname{mgh}}{\mathrm{ms}}$
Hence after we put the values, we can write them as:
$\Rightarrow \theta=\dfrac{8.82}{0.018 \times 80} \times$0.018
(heat capacity of 1 mole of water $(\mathrm{s})=80 \mathrm{J} / \mathrm{K})$
So, we get the value as:
$\Rightarrow \theta=0.11^{\circ} \mathrm{C}$
Hence the difference of temperature at the top and the bottom of the fall is $0.11^{\circ} \mathrm{C}$.

From the above we can say that the options B and C are correct.

Note: It is known that heat energy is the result of the movement of tiny particles called atoms, molecules or ions in solids, liquids and gases. Heat energy can be transferred from one object to another. The transfer or flow due to the difference in temperature between the two objects is called heat. Heat is the form of energy that is transferred between systems or objects with different temperatures (flowing from the high-temperature system to the low-temperature system). Also referred to as heat energy or thermal energy. Heat is typically measured in Btu, calories or joules. Thermal energy (also called heat energy) is produced when a rise in temperature causes atoms and molecules to move faster and collide with each other. The energy that comes from the temperature of the heated substance is called thermal energy.