A water level is maintained up to a fixed height H in a cylindrical vessel. The vessel rests on a horizontal plane. At what height above the bottom should a hole be made in the vessel such that the water stream emerging from the hole strikes the horizontal plane at the maximum distance from the vessel, as shown in the figure?

Step-by-step solution

Let the hole be made at a height y from the bottom of the vessel. The depth of the hole from the free surface of the water is H - y . By Torricelli's theorem, the velocity of efflux is: v = 2g(H - y) The time taken by the water stream to reach the horizontal plane is: t = 2y g The horizontal range x of the water stream is given by: x = vt Substituting the expressions for v and t , we get: x = 2g(H - y) 2y g = 2 y(H - y) To maximize the horizontal range x , the quantity inside the square root, i.e., y(H - y) , must