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A helium balloon has volume V0 and temperature T0 at sea level where the pressure is P0 and the air density is ρ0. The balloon is allowed to float up in the air to altitude y where the temperature is T1. (a) Show that the volume occupied by the balloon is then V = V0(T1/T0)e+cy where c = ρ0g/P0 = 1.25 × 10−4 m−1. (b) Show that the buoyant force does not depend on altitude y. Assume that the skin of the balloon maintains the helium pressure at a constant factor of 1.05 times greater than the outside pressure. [Hint: Assume that the pressure change with altitude is P = P0e−cy, as in Example 13–5, Chapter 13.]
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