The ideal gas equation:
The density of an ideal gas:
Weight of the hot air = mhg = ρhV1g
Total weight, W, of the ballon including the hot air inside:
W = mbg + ρhV1g
The Buoyant force on the balloon is equal to the weight of cold air displaced:
Hot air balloons float in the air because of the difference in density between cold and hot air. Consider a balloon in which the mass of the pilot basket together with the mass of the balloon fabric and other equipment is mb. The volume of the hot air inside the balloon is V1 and the volume of the basket, fabric, and other equipment is V2. The absolute temperature of the hot air at the bottom of the balloon is Th (where Th > Tc). The balloon is open at the bottom, so that the pressure inside and outside the balloon is the same. Assume that we can treat air as an ideal gas. Use g for the magnitude of the acceleration due to gravity. For the balloon to float, what is the minimum temperature Tmin of the hot air inside in terms of the variables given?
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Based on our data, we think this problem is relevant for Professor Beving's class at UCR.