Water level is maintained in a cylindrical vessel upto a fixed height $H$ . The vessel is kept on a horizontal plane. At what height above the bottom should a hole be made in the vessel, so that the water stream coming out of the hole strikes the horizontal plane of the greatest distance from the vessel?

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The velocity with which water will came out v = 2 g ( H − h ) ​ The time of flight of water ejected from the hole. t ∴ x ​ = g 2 h ​ ​ = time of flight = v × t = 2 g ( H − h ) ​ × g 2 h ​ ​ = 4 ( H h − h 2 ) ​ ​ For x to be maximum ⇒ ⇒ ​ d h d x ​ = 0 d d h 4 ( H h − h 2 ) ​ ​ = 0 0 = H − 2 h ⇒ h = 2 H ​ ​

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