Chapter 17, Problem 88GP

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.]