Question

A cylinder containing water stands on a table of height H. A small hole is punched in the side of the cylinder at its base. The stream of water strikes the ground at a horizontal distance R from the table. Then the depth of water in the cylinder is:
\[\begin{align}
  & \left( A \right)H \\
 & \left( B \right)R \\
 & \left( C \right)\sqrt{RH} \\
 & \left( D \right)\dfrac{{{R}^{2}}}{4H} \\
\end{align}\]

Answer 1

Hint: These types of problems are quite easy to solve once we understand the underlying concepts behind the sums. This particular problem requires some prerequisite knowledge of physics as well some algebraic concepts of mathematics. We need to be through with the chapter of kinematics to be able to solve the problem efficiently. Here in this problem we need to assume that the height of the cylinder is equal to ‘h’, and we need to recall from the concepts of fluid dynamics that the velocity of the stream flow from the bottom of a cylinder of height ‘h’ is given by,
\[v=\sqrt{2gh}\] . Using this relation we apply the concept that the time required by the stream of water to cover the horizontal distance and the vertical distance will be equal.

Complete step-by-step solution:
Now we start off with the solution to the problem by writing that, the time required for the stream of water to travel the horizontal distance ‘R’ is given by,
\[\begin{align}
  & t=\dfrac{R}{v} \\
 & \Rightarrow t=\dfrac{R}{\sqrt{2gh}} \\
\end{align}\]
Now we find the time required for the stream to cover the vertical distance ‘H’. It can be found out by applying the 1-D equation of motion,
\[H=ut+\dfrac{1}{2}g{{t}^{2}}\]
Here in this problem the initial velocity \[u=0\] . Thus the equation transforms to,
\[H=\dfrac{1}{2}g{{t}^{2}}\]
Finding the value of ‘t’ from this equation we get,
\[t=\sqrt{\dfrac{2H}{g}}\]
Equating this value of ‘t’ with the previous found out value we get,
\[\sqrt{\dfrac{2H}{g}}=\dfrac{R}{\sqrt{2gh}}\]
From this we need to find out the value of ‘h’ or the height of the cylinder. We evaluate it as,
\[h=\dfrac{{{R}^{2}}}{4H}\]
Thus the answer to our problem is option D.

Note: For such types of problems we need to be fluent in physics as well as mathematics. We need to know some of the basic equations of motion, fluid mechanics as well as the concepts of algebra. In this problem we need to assume the height of the cylinder and then apply the equations of motion and find out the time required for the stream of water to cover the horizontal and vertical distance. Both of these times will be equal and hence we can equate these two values to find out our required answer.