Question

The height of water level in a tank is H. The range of water stream coming out of a hole at depth $\dfrac{H}{4}$ from the upper water level will be:
$\begin{aligned} & A)\dfrac{\sqrt{3}}{2}H \\\ & B)\dfrac{2H}{\sqrt{3}} \\\ & C)\dfrac{H}{\sqrt{3}} \\\ & D)H\sqrt{3} \\\ \end{aligned}$

Answer 1

In this problem, the concepts of both fluid mechanics and horizontal motion of a projectile are used together. The Torricelli’s theorem or the Torricelli’s principle (derived from Bernoulli's theorem) which is used to calculate the velocity of efflux is applied in this case to find the range of the water stream coming out of the tank.
Formula Used:
$R$ $=2\sqrt{h(H-h)}$

Complete answer:
The velocity of efflux given by the Torricelli’s principle is given as follow:
$v=\sqrt{2gh}$ $......(1)$
The time taken for the liquid to reach the base of the vessel is calculated as follows:
$t=\sqrt{\dfrac{2(H-h)}{g}}$ $......(2)$
Here, h is the depth at which the hole is made from the surface of the liquid in the vessel.
g is the acceleration due to gravity.
H is the height vessel where the liquid is kept.
$\begin{aligned} & (1)\times (2)\Rightarrow \\\ & v\times t=\sqrt{2gH}\times \sqrt{\dfrac{2(H-h)}{g}} \\\ & \\\ \end{aligned}$
$=2\sqrt{h(H-h)}$
This equation gives us the range of the water stream coming out of the water vessel.
Therefore, the range is given by:
$R$ =$2\sqrt{h(H-h)}$

Fig 1: Cylindrical vessel showing height H, and depth H/4
Given,
Height of the water tank is H.
Hole is at a depth of$\dfrac{H}{4}$.
To find the range of water stream coming out of the tank,
$\begin{aligned} & R=2\sqrt{\dfrac{H}{4}(H-\dfrac{H}{4}}) \\\ & \\\ \end{aligned}$
$\Rightarrow R=\dfrac{\sqrt{3}}{2}H$

The range of water stream coming out is given by $(A)\dfrac{\sqrt{3}}{2}H$.

Additional Information:
Torricelli's equation or theorem relates the speed of fluid flowing out of a vessel. It is known as the velocity of efflux. The scientist who discovered this law was Evangelista Torricelli.

Note:
Care must be taken while noting the height at which the hole is located. The height in the equation is actually the depth of the hole from the water level. Instead if the height is given from the bottom of the vessel, it must be subtracted from the total vessel height to obtain the value of h.