A cylindrical vessel filled with water to a hight H. A vessel has two small holes in the side, from which water iins rushing out horizontal and the two steams stike the ground at the same poin, if the lower hole Q Q is h height above the ground, then the height of hole P above the ground will be

Options

A. 1
B. 2
C. 3
D. 4

Explanation

Suppose there is a hole in the cylinder at depth y below water level

Velocity of efflux `v = sqrt(2gy)`
time taken by water to reach on the plane PQ will be
` t = sqrt( (2(h/2 + (h-y) ))/(g) ) = sqrt( (3h - 2y)/(g))`
` therefore ` horizontal distance x travelled by the liquid is
` x = vt = sqrt(2gy) sqrt( (3h - 2y)/(g))`
`x = sqrt(2y(3h - 2y) )`
for x to be maximum `(dx)/(dy) = 0`
`(1)/(2sqrt(2y(3h - 2y) )) xx 2 (3h - 4y) = 0`
`3h - 4y = 0 , therefore y = (3h)/(4)`
Hence , x will be maximum at `y = (3h)/(4) = (h - h/4)`
where hole number 2 is present.