PROBLEM 3 Advertised is a small toy that will send up a signal flare with a parachute. Upon examination, we found that it is simply a metal tube 1.5 m long and 400 mm2 in cross-sectional area. A plug, which contains the parachute, fits into the tube and a trigger holds it in place. At this position, the bottom edge of the plug is 0.7 m from the bottom. The plug weighs 0.2 kg. To operate the toy, the air below the plug is pumped up to a pressure of 3 bar with a hand pump, and then the trigger is released, allowing the plug to fly out the top. The ambient temperature is 298 K, and the constant- volume heat capacity of air is 2.5 R, where R is the gas constant. (a) Assuming the entire device operates isothermally at 298 K, what is the maximum height (measured from the ground) that the plug can reach? Hint: To get the maximum possible height, you may assume that the gas expansion is quasi-static, that is, the gas remains in an equilibrium state at all times. (b) Although the tube is made of metal, a good conductor of heat, the expansion is so quick that there may not be enough time for much heat transfer. Repeat the calculation of Part (a), this time assuming the expansion of the compressed air is adiabatic instead of isothermal. (c) The real process will not be isothermal or adiabatic, but somewhere in between. A way to describe such a real process is to assume a polytropic" process, for which PV = constant at all intermediate states during the expansion, where & is an empirical (i.e., experimentally measured) constant called the polytropic index. We found that 8 = 1.25. Repeat the calculation of the maximum height under this assumption. Also find the heat transfer into the compressed air up to the time point when the plug leaves the tube