Question

The velocity of a liquid coming out of a hole in the tank wall is:
\[\left( A \right)\] More if the hole is near the upper end.
$\left( B \right)$ More if the hole is at the center of the wall.
$\left( C \right)$ More if hole is near the bottom.
$\left( D \right)$ Velocity of flow does not depend upon the position of the hole.

Answer 1

Hint: In this question use the property that the speed of an object falling freely from a height h is same as the speed of efflux of a fluid through a sharp-edged hole at the bottom of a tank filled to a depth h, remember that both the height and the depth should be same so use these concepts to reach the solution of the question.

Complete Step-by-Step solution:
Let us consider an object is placed at a height h now we have to free fall this object so its initial velocity becomes zero and let its final velocity be (v). When it reaches the ground and we all know the acceleration of gravity is acting downwards therefore it is (a = g) m/$s^2$
So from third equation of motion we have,
${v^2} = {u^2} + 2as$..................... (1), where v = final velocity, u = initial velocity, a = acceleration due to gravity and s = height of the particle.
Now substitute the values in equation (1) we have,
$ \Rightarrow {v^2} = {0^2} + 2\left( g \right)h$
$ \Rightarrow v = \sqrt {2gh} $
So the speed of the object falls freely from a height h is $\sqrt {2gh} $.
Now according to Torricelli's theorem the speed of the efflux (i.e. material that is flowing) from the bottom of the tank having depth (h) (i.e. having a sharp hole at the bottom of the tank) is equal to the speed of the object having free fall from a height h.
Remember that height h of an object should be equal to the depth of the tank.
Therefore we use the h symbol for both the cases.
So the speed of the efflux from the bottom of the tank having depth h and having a sharp hole at the bottom of the tank is $\sqrt {2gh} $.
Therefore as soon as the depth of the tank decreased by making a hole not at the bottom of the tank but at the middle of the tank or at the upper part of the tank the velocity starts decreasing according to the above formula.
Therefore the velocity of liquid coming out from a hole is greater when the hole is near the bottom.
So this is the required answer.
Hence option (C) is the correct answer.

Note – Whenever we face such types of question the key concept we have to remember is Torricelli's theorem which is stated above so according to this theorem the speed of the liquid having a hole at the bottom of the tank having depth h is given as $\sqrt {2gh} $ therefore, if depth decreased the velocity is also decreased so we have to maximize the depth this is only possible if the hole is at bottom of the tank.