Question

A cylindrical vessel is filled with a liquid up to a height $H$. A small hole is made in the vessel at a distance $y$ below the liquid surface as shown in figure. The liquid emerging from the hole strikes the ground at distance $X$.

• A. $X$ remains same if the hole is at depth $y$ or $H-y$.
• B. $X$ is maximum for $y=\frac{H}{2}$.
• C. both (a) and (b) are correct.
• D. both (a) and (b) are wrong.

Answer 1

The correct option is C both (a) and (b) are correct.

We know, range of efflux from the hole is given by -
$X=2\sqrt{y(H-y)}$
If hole is made at depth $y^{{}^{\prime }}=H-y$, then
$X^{{}^{\prime }}=2\sqrt{(H-y)[H-(H-y\left)\right]}$
$=2\sqrt{y(H-y)}$
$=X$
Hence, range will remain same.


Now, $\frac{dX}{dy}=\frac{1}{2}(yH-y^2{)}^{-1/2}(H-2y)$
For maximum value of $X$, $\frac{dX}{dy}=0$
$\Rightarrow \frac{H-2y}{2(yH-y^2{)}^{1/2}}=0$
[Since denomenator can't be zero, $y\ne 0\&y\ne H$]
$\Rightarrow y=\frac{H}{2}$
Both statements are true.
So, option $\text{C}$ is correct.