Spring 2010 Solutions - FSU Physics Department

Problem 9The Jet d'Eau, or water-jet, is a large fountain in Geneva, Switzerland,and is one of the city's most famous landmarks. Situated at the pointwhere Lake Geneva empties into the Rhone River, it is visiblethroughout the city and from the air, and is one of the largest fountainsin the world. It is designed to spray a column of water 140 meters intothe air and has a nozzle 5 cm in diameter at ground (or lake surface)level. The water pump is 6 meters below the ground level, and the pipeto the nozzle has a diameter of 15 cm. P atm = 101kPa. (Neglect theviscosity of the water and the air drag on the rising column of water inthe air.)(a) What is the velocity of the water leaving the nozzle (at ground level)?(b) What is the flow rate of the water just as it leaves the nozzle (in liters/sec)?(c) What pump pressure is required?Solution:…one of the largest fountains in the world. It is designed to spray a column of water 140meters into the air and has a nozzle 5 cm in diameter at ground (or lake surface) level.The water pump is 6 meters below the ground level, and the pipe to the nozzle has adiameter of 15 cm. P atm = 101kPa. (Neglect the viscosity of the water and the air drag onthe rising column of water in the air.)(a) What is the velocity of the water leaving the nozzle (at ground level)?We note that it is useful for the solution of this problem to obtain the state of the fluidat four distinct points along its path” (1) at the water pump, (2) just before the nozzle,(3) just outside the nozzle, and (4) at the top of the column of water. To solve this weneed a relation between the velocity of an incompressible fluid, its pressure at twoplaces, and the height difference between the two places. The relation is theBernoulli equation, which is a statement of conservation of energy in a fluid.1 2 1 2P1+ ρgy1+ 2ρv1 = P2 + ρgy2+2ρv2We also need the equation for continuity of the fluid, which is a statement of the lackof sources or sinks of fluid (or equivalently, of conservation of momentum in thefluid),vA= vA1 1 2 2where, for example, A 1 is the cross sectional area of the pipe at position (1).The parameters at each point along the fluid flow path:(1) At the water pump we have P 1 , y 1 = -6.0 m, v 1(2) Just before the nozzle we have P 2 , y 2 = 0, v 2(3) Just after the nozzle we have P 3 = P atm , y 3 = 0, v 3 = 9v 2 = 9v 1 , because the crosssectional area went down by a factor of nine going through the nozzle.(4) At the top of the column of water we have P 4 = P atm , y 4 = 140 m, v 4 = 0.