Question

Water level is maintained in a cylindrical vessel up to a fixed height H. The vessel is kept on a horizontal plane. At what height above the bottom should a hole be made in the vessel so that the water stream coming out of the hole strikes the horizontal plane at the greatest distance from the vessel.

Answer 1

It is given that H is the height of the cylindrical vessel.

Now, let h be the height of the hole from the surface of the tank.
The velocity of water \[\left( v \right)\] is given by
\[v = \sqrt{2g(H - h)}\]
Also, let t be the time of flight. 
Now, 

\[t = \sqrt{\frac{2h}{g}}\]

Let x be the maximum horizontal distance . 

\[ \therefore x = v \times t\]

\[ = \sqrt{2g(H - h)} \times \sqrt{\frac{2h}{g}}\]

\[ = \sqrt{4(Hh - h^2 )}\]

For x to be maximum,

\[\frac{d}{dh}\left( Hh - h^2 \right) = 0\]

\[ \Rightarrow 0 = H - 2h\]

\[ \Rightarrow h = \frac{H}{2}\]