A very wide tank is filled with water to a depth above its flat bottom
PHYSICS · Multi-part · CBSE · Physics · Mechanical Properties of Fluids · Mechanical Properties of Fluids
Problem
A very wide tank is filled with water to a depth H above its flat bottom. A narrow, sharp-edged hole of negligible area compared with the tank’s cross-section is drilled horizontally through the side wall at a distance y below the water surface, as shown in the description below.
Water leaves the hole horizontally, travels through the air, and strikes the level ground a horizontal distance R from the base of the tank (ground and tank bottom are at the same vertical level). Assume that the water level in the tank remains essentially constant during the motion of a fluid element and that air resistance on the jet is negligible.
(a) Derive an algebraic expression for the initial speed v0 of the water as it leaves the hole in terms of g, y, and any other necessary quantities.
(b) Determine the time tf it takes for the water to fall from the hole to the ground in terms of y, H, and g.
(c) Obtain an expression for the horizontal range R of the jet as a function of y and H.
(d) Using algebraic reasoning (without calculus), show that R is maximized when the hole is placed halfway between the water surface and the bottom of the tank, that is, when y=2H. State the corresponding maximum value of R in terms of H and g.