Question

A water tank of height 10m, completely filled with water is placed on a level ground. It has two holes one at 3m and the other at 7m from its base. The water ejecting from:
A. Both the holes will fall at the same spot
B. Upper hole will fall farther than that from the lower hole
C. Upper hole will fall closer than that from the lower hole
D. More information is required

Answer 1

Hint:The question is based on the application of Bernoulli’s equation: speed of efflux. Determine the time required for the water ejecting from the hole to hit the ground using the kinematic equation in the vertical direction. Using the kinematic equation in the horizontal direction of the water flow, you will be able to calculate the range of the ejecting water stream.

Formula used:
\[s = ut + \dfrac{1}{2}a{t^2}\]
where, s is the distance, u is the initial velocity, t is the time and a is the acceleration.
Bernoulli’s equation, \[v = \sqrt {2gh} \]
where, g is the acceleration due to gravity and h is the height of the orifice from the top of the free surface of the liquid.

Complete step by step answer:
As we can see the question is based on the application of Bernoulli’s equation: speed of efflux. We can determine the range of the water ejecting from an orifice using the kinematic equation in the horizontal direction of the water flow. Thus, we have to determine the time required for the water ejecting from the hole to hit the ground. So, we will use the kinematic equation in the vertical direction of motion of the water to determine the time as follows,
\[h = ut + \dfrac{1}{2}g{t^2}\]
Here, h is the height of the hole (orifice) from the base, u is the initial velocity of the water ejecting from the hole, g is the acceleration due to gravity and t is the time.
Since the initial velocity of the ejecting water is zero, the above equation becomes,
\[h = \dfrac{1}{2}g{t^2}\]
\[ \Rightarrow t = \sqrt {\dfrac{{2h}}{g}} \] …… (1)
Now, the speed of the efflux is can be expressed as,
\[v = \sqrt {2gh} \]
But, in the above equation, h is the height from the free surface of the water tank that is from the top of the water surface. Thus, we can express the height from the base of the tank as,
\[v = \sqrt {2g\left( {10 - h} \right)} \] …… (2)
Here, 10m is the height of the tank as given in the question.
Let us express the horizontal range of the water using the kinematic equation in the horizontal direction as,
\[s = vt\]
Using equation (1) and (2) in the above equation, we get,
\[s = \left( {\sqrt {2g\left( {10 - h} \right)} } \right)\left( {\sqrt {\dfrac{{2h}}{g}} } \right)\]
\[ \Rightarrow s = 2\sqrt {\left( {10h - {h^2}} \right)} \]
Now, let us calculate the range of the water flow for the hole drilled at 3m from the base as,
\[s = 2\sqrt {\left( {10\left( 3 \right) - {{\left( 3 \right)}^2}} \right)} \]
\[ \Rightarrow s = 9.16\,{\text{m}}\]
Also, the range of the water flow for the hole drilled at 7m from the base is,
\[s = 2\sqrt {\left( {10\left( 7 \right) - {{\left( 7 \right)}^2}} \right)} \]
\[ \therefore s = 9.16\,{\text{m}}\]
Therefore, the water ejecting from both the holes will fall at the same spot.

So, the correct answer is option A.

Note: The Bernoulli’s equation is derived from the law of conservation of energy. Thus, in case you don’t remember Bernoulli’s equation of efflux, you can use the law of conservation of energy where the potential energy is converted into the kinetic energy. Note that in Bernoulli’s equation, the height h is from the top of the free surface of the water and not from the base.