Consider a water jar of radius $R$ that has water filled up to height $H$ and is kept on a stand of height $h$ (see figure) . Through a hole of radius $r$ $(r << R)$ at its bottom, the water leaks out and the stream of water coming down towards the ground has a shape like a funnel as shown in the figure. If the radius of the cross-section of water stream when it hits the ground is $x.$ Then
JEE MAIN 2016, Diffcult
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According to Bernoulli's principle,
$\frac{1}{2}\rho v_1^2 + \rho gh = \frac{1}{2}\rho v_2^2$
$v_1^2 + 2gh = v_2^2$
$2gH + 2gh = v_2^2\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( i \right)$
${a_1}{v_1} = {a_2}{v_2}$
$\pi {r^2}\sqrt {2gh} = \pi {x^2}{v_2}$
$\frac{{{r^2}}}{{{x^2}}}\sqrt {2gh} = {v_2}$
Substituting the value of $v_2$ in equation $(i)$
$2gH + 2gh = \frac{{{r^4}}}{{{x^4}}}2gh\,\,or,\,\,x = r{\left[ {\frac{H}{{H + h}}} \right]^{\frac{1}{4}}}$
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