Question

Water is filled in a tank upto $3\text{m}$ height. The base of the tank is at height $1\text{m}$ above the ground. What should be the height of a hole made in it, so that water can be sprayed upto maximum horizontal distance on the ground?

• A. $3\text{m}$ from ground
• B. $1.5\text{m}$ from ground
• C. $2.5\text{m}$ from ground
• D. $2\text{m}$ from ground

Answer 1

The correct option is D $2\text{m}$ from ground

Let us suppose, height of the hole from the ground is $x$.
According to question -


We know that, velocity of efflux of water, $v=\sqrt{2gh}$
$\Rightarrow v=\sqrt{2g(4-x)}$ ..........(1)
$[$ from figure $]$
Considering the vertical motion of water,
$x=\frac{1}{2}g{t}^2$
$t=\sqrt{\frac{2x}{g}}$ ..........(2)
$[$ time taken by water to reach ground $]$
Horizontal range of water, $R=vt$
$\Rightarrow R=\sqrt{2g(4-x)}\times \sqrt{\frac{2x}{g}}$
$[$ from (1) and (2) $]$
$\Rightarrow R=\sqrt{16x-4x^2}$
Now, $\frac{dR}{dx}=\frac{1}{2}(16x-4x^2{)}^{\frac{-1}{2}}.(16-8x)$
For maximum range, $\frac{dR}{dx}=0$
$\Rightarrow \frac{16-8x}{2\times \sqrt{(16x-4x^2)}}=0$
$[x\ne 0$ and $x\ne 4$ as denominator cannot be equal to zero $]$
$\Rightarrow 16-8x=0$
$\Rightarrow x=2\text{m}$