Question

A cylinder drum, open at the top, contains 15 litres of water. It drains out through a small opening at the bottom. 5 litres of water comes out in time, ${{t}_{1}}$, the next 5litres in further time ${{t}_{2}}$ and the last 5 litres in further time ${{t}_{3}}$.Then
$\begin{aligned} & (A){{t}_{1}}< {{t}_{2}}< {{t}_{3}} \\\ & (B){{t}_{1}}> {{t}_{2}}> {{t}_{3}} \\\ & (C){{t}_{1}}={{t}_{2}}={{t}_{3}} \\\ & (D){{t}_{1}}> {{t}_{2}}={{t}_{3}} \\\ \end{aligned}$

Answer 1

Use Torricelli's theorem. According to Torricelli’s theorem velocity of efflux is directly proportional to the height of water flow. When the velocity of efflux decreases, height also decreases. That is, the speed of liquid flowing out under the gravity is directly proportional to the height of water level.

Complete answer:
Velocity of efflux, $v=\sqrt{2gh}$
where h is the height of the liquid, g is the acceleration due to gravity.
According to Torricelli’s theorem velocity of efflux is directly proportional to the height of water flow. When the velocity of efflux decreases, height also decreases. That is, the speed of liquid flowing out under the gravity is directly proportional to the height of water level.
The velocity is proportional to the height of water coming out. The water level decreases with increase in time. As water flows out, the height of the water level also decreases. This results in a decrease in water flow. That is, the rate of water level decreases with time.
Therefore, ${{t}_{1}}< {{t}_{2}}< {{t}_{3}}$

So, the correct answer is “Option A”.

Note:
According to Torricelli’s theorem velocity of efflux is directly proportional to the height of water flow. When the velocity of efflux decreases, height also decreases. That is, the speed of liquid flowing out under the gravity is directly proportional to the height of water level. The water level decreases with increase in time. As water flows out, the height of the water level also decreases. This results in a decrease in water flow. That is, the rate of water level decreases with time.