Question

The Jog falls in Karnataka state are nearly 20 m high. 2000 tonnes of water falls from it in a minute. Calculate the equivalent power if all this energy can be utilized ($g=10m{{s}^{-2}}$ ):
(A). $6.67\times {{10}^{5}}W$
(B). $6.67\times {{10}^{6}}W$
(C). $6.67\times {{10}^{8}}W$
(D). $6.67\times {{10}^{-6}}W$

Answer 1

Hint: To solve this question potential energy of the given amount of water at the height of the fall of 20 m. this potential energy will be equal to the work done by the water to fall. Again, the power of the water can be found by dividing the work done by the time in which the water falls below. You will find your answer.

Complete step by step answer:
We are given in the question that the height of the fall is $h=20m$ .
Mass of the water $m=2000\text{ tonnes = 2000}\times \text{1000 kg = 2}\times \text{1}{{\text{0}}^{6}}kg$
The time at which the given mass falls from the fall is $t=1\text{ minute }=\text{ }60\text{ seconds}$
Now, power can be defined as the work done per unit time. The work done when the water falls from the fall is given by the potential energy of the water above the fall.
So, work done $=mgh$
Where, m is the mass of water, g is the acceleration due to gravity and h is the height of the fall.
Work done $\begin{aligned} & =mgh \\\ & =2\times {{10}^{6}}\times 10\times 20 \\\ & =4\times {{10}^{8}}J \\\ \end{aligned}$
So, we can define the power as,
Power $\begin{aligned} & =\dfrac{\text{work done}}{\text{time}} \\\ & =\dfrac{4\times {{10}^{8}}}{60} \\\ & =6.67\times {{10}^{6}}\text{ watt} \\\ \end{aligned}$
So, the equivalent power obtained if all the energy of the fall is utilized it will be $6.67\times {{10}^{6}}W$
The correct option is (B)

Note: With the help of law of conservation energy we can solve this question. The potential energy of the water above the fall will transform into the kinetic energy of the water when it starts falling. At the bottom of the fall the potential energy of the water will be zero and the kinetic energy of the water will be equal to the potential energy of water above the fall. That’s why we can say that work done will be equal to the potential energy of the water above the fall.