A container of large uniform cross sectional area `A`, resting on horizontal surface, holds two immiscible non viscous and incompressible liquids of density `d` and `2d`, each of height `H/2` as shown in the figure. The lower density liquid is open to the atmosphere having pressure `P_(0)`. A tiny hole of area `(s(sltltA)` is punched on the vertical side of the container at a height `h(hltH/2)`. Determine

a.the initial speed of efflux of the liquid at the hole
b. the horizontal disance `x` travelled by the liquid initially
c. the height `h_m` at which at the hole should be punched so that the liquid travels the maximum distance `x_(m)` initially. also calculate `x_(m)` (neglect air resistance in calculations).

Explanation

(a) Atmospheric pressure : `P_0`
Applying Bernoulli's theorem between `E` and `F`
`P_0 + dg .(H)/(2) + 2dg((H)/(2) -h) = P_0 + (1)/(2). 2d v^2`
`(gH)/(2) + 2g((H)/(2) -h) = v^2`
`v^2 = (3 gH)/(2) - 2gh =(g)/(2) (3H - 4h)`
`v = sqrt((g)/(2) (3H - 4h))`
(b) `F` to `G` (projectile motion)
`h = (1)/(2) g t^2 rArr t = sqrt((2h)/(g))`
`x = vt = sqrt((g)/(2)(3 H - 4h).(2h)/(g)) = sqrt(h(3 H -4h))`
( c) For maximum value of `x, (dx)/(dh)` or `(dx^2)/(dh) = 0`
`(d(x^2))/(dh) = (d)/(dh) (3h H - 4h^2) = 3H - 8h = 0`
`h = (3H)/(8)`
`x_m = sqrt((3H)/(8)(3H - (3H)/(2))) = (3H)/(4)`.