How many cubic meters of helium are required to lift a...

How many cubic meters of helium are required to lift a balloon with a 400-kg payload to a height of 8 000 m? Take $\rho_{\mathrm{He}}=0.179 \mathrm{kg} / \mathrm{m}^{3} .$ Assume the balloon maintains a constant volume and the density of air decreases with the altitude $z$ according to the expression $\rho_{\text { air }}=\rho_{0} e^{-z / 8000}$ , where $z$ is in meters and $\rho_{0}=1.20 \mathrm{kg} / \mathrm{m}^{3}$ is the density of air at sea level.

Key Concepts

Constant Volume Balloon Assumption

Assuming the balloon maintains a constant volume simplifies the problem since the volume (and thus the amount of helium) remains unchanged during ascent. This means that while the external air density decreases with altitude, the properties of the helium inside do not vary, simplifying the calculation of net lift.

Payload and Net Lift Calculation

To achieve lift, the net upward force provided by the buoyant force must exceed the weight of the payload. This involves computing the difference in density between the air and the helium and ensuring that the displaced air's weight is sufficient to counterbalance the combined weight of the balloon's payload and the helium.

Archimedes' Principle (Buoyant Force)

This principle states that a body immersed in a fluid experiences an upward force equal to the weight of the fluid it displaces. In the context of a balloon, the lift is generated by the difference between the weight of the displaced air and the weight of the helium filling the balloon.

Density Variation with Altitude

The density of the atmosphere decreases with increasing altitude, often modeled using an exponential decay function. This variation is crucial in determining the effective buoyant force at different heights, as the lift depends on the difference between the local air density and the balloon gas density.

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