Since the end of World War II, various claims have appeared in the popular aviation literature of instances where powerful propeller-driven fighter airplanes from that period have broken the speed of sound in a vertical, power-on dive. The purpose of this problem is to show that such an event is technically not possible. Consider, for example, the Grumman F6F-3 Hellcat, a typical fighter from World War II. For this airplane the zero-lift drag coefficient (at low speeds) is 0.0211, the wing planform area is 334 ft2, and the gross weight is 12,441 lb. It is powered by a Pratt and Whitney R-2800 reciprocating engine that, with supercharging to an altitude of 17,500 ft, produces 1500 horsepower. Consider this airplane in a full-power vertical dive at (a) 30,000 ft and then (b) 20,000 ft. Prove that at these two altitudes the airplane cannot reach Mach 1.
Note: The aerodynamic characteristics of this airplane at Mach 1 have not been measured. So you will have to make some reasonable assumptions. For example, what is the zero-lift drag coefficient at Mach 1? As an estimate, we can obtain from NACA TR 916 a zero-lift drag coefficient for the North American P-51 Mustang, which, when extrapolated to Mach 1, shows an increase of 7.5 over its low-speed value. For the more blunt configuration of the F6F, let us assume that CD,0 (at M = 1) is 10 times larger than CD,0 (low speed). Also, at Mach 1 the propeller efficiency would be almost zero (indeed, the propeller might even be producing a net drag rather than any thrust). To be conservative, let us assume the propeller efficiency at Mach 1 to be 0.3.
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