Final answer:
The question involves calculating the required gauge pressure for water to exit a tapered hose at a specified velocity, considering the elevation difference between the hose ends.
Step-by-step explanation:
The question you asked pertains to fluid dynamics within the context of Physics. When discussing hose diameters and the required pressures for a fluid (like water) to exit at a certain velocity, we're dealing with aspects of Bernoulli's equation and concepts related to pressure and flow rate. If a pipe is tapered such that its large end has a diameter that is twice as large as the small end, and you want water to emerge from the small end at a specific speed (in this case, 12 m/s) while also elevating the small end by 8 m above the large one, you must calculate the gauge pressure — the difference between the pressure at the large end and the small end. This calculation involves considering gravitational potential energy, kinetic energy of the fluid, and the continuity equation which relates the flow rates and cross-sectional areas at both ends of the hose.
Given that the pressure P2 in the nozzle must be atmospheric because the water emerges into the atmosphere, the absolute pressure in the hose will undoubtedly be greater than in the nozzle, since the velocity (v) is greater in the nozzle as a result of the conservation of mass for incompressible fluids. The required gauge pressure therefore accounts for the kinetic energy of fluid moving at this speed, the difference in gravitational potential energy due to the height difference, and the atmospheric pressure at the exit point.