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Q.

A container of large uniform cross – sectional area holds two immiscible, non – viscous and incompressible liquids of densities ρ  and 4ρ , each of height H2 . Determine the position where, a small hole has to be punched, so that the heavier liquid comes out with a maximum range R initially.

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a

At a height H4  from the bottom

b

At a height 54H  from the bottom

c

At a height 516H  from the bottom

d

At a height 718H  from the bottom

answer is C.

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Detailed Solution

On applying Bernoulli’s theorem between points (1) and (2), we havep0+ρgH2+4ρg(H2−h)=p0+124ρu2ρgH2+2ρgH−4ρgh=2ρu25ρgH2−4ρgh=2ρu2 ⇒u2=[54gH−2gh]⇒u2=g4[5H−8h]    ⇒u=g25H−8h   t=2hgR=ut=g25H−8h.2hg=125Hh−8h2For R to be maximum,dRdh=0    dRdh=12.125Hh−8h2(5H−16h)=0⇒5H−16h=0 ⇒h=516H
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