Hot air balloons float in the air because of the difference in density between cold and hot air. In this problem, you will estimate the minimum temperature the gas inside the balloon needs to be, for it to take off. To do this, use the following variables and make these assumptions:
The combined weight of the pilot basket together with that of the balloon fabric and other equipment is W.
The volume of the hot air inside the balloon when it is inflated is V.
The absolute temperature of the hot air at the bottom of the balloon is T_h (where T_h is greater than T_c ).
The absolute temperature of the cold air outside the balloon is T_c and its density is p_c .
The balloon is open at the bottom, so that the pressure inside and outside the balloon is the same.
As always, treat air as an ideal gas.
Use g for the magnitude of the acceleration due to gravity.

Part A
What is the density p_h of hot air inside the balloon? Assume that this density is uniform throughout the balloon.
Express the density in terms of T_h, T_c, and p_c.

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  1. p_h = p_c * (T_c / T_h)

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