Halliday, Resnick & Walker — Fluids: Problem 48

48 A pitot tube (Fig. 14-41) is used to determine the air speed of an airplane. It consists of an outer tube with a number of small holes B (four are shown) that allow air into the tube; that tube is connected to one arm of a U-tube. The other arm of the U-tube is connected to hole A at the front end of the device, which points in the direction the plane is headed. At A the air becomes stagnant so that \(v_A = 0\). At B, however, the speed of the air presumably equals the airspeed \(v\) of the plane. (a) Use Bernoulli’s equation to show that \(v = \sqrt{\frac{2\rho gh}{\rho_{air}}}\), where \(\rho\) is the density of the liquid in the U-tube and \(h\) is the difference in the liquid levels in that tube. (b) Suppose that the tube contains alcohol and the level difference \(h\) is 20.0 cm. What is the plane’s speed relative to the air? The density of the air is \(1.03\text{ kg/m}^3\) and that of alcohol is \(810\text{ kg/m}^3\).