Why does a hot-air balloon rise in the atmosphere?

• A. Because the heated air inside the balloon increases the pressure, which pushes the balloon upward.
• B. Because the balloon material expands when heated, creating lift.
• C. Because the hot air inside the balloon is heavier than the cooler air outside, causing the balloon to rise.
• D. Because the air inside the balloon is heated, its density $\rho$ decreases, making the weight of the displaced air greater than the weight of the balloon, resulting in a net upward buoyant force.

Answer: (D)

Verified step by step guidance

1. Understand that a hot-air balloon rises due to the principle of buoyancy, which depends on the difference in density between the air inside the balloon and the surrounding air.
2. Recall that when air is heated, its temperature increases, causing the air molecules to move faster and spread apart, which decreases the air's density inside the balloon. This can be expressed as \(\rho = \frac{m}{V}\), where \(\rho\) is density, \(m\) is mass, and \(V\) is volume; heating increases \(V\) for the same \(m\), so \(\rho\) decreases.
3. Apply Archimedes' principle, which states that the buoyant force on an object immersed in a fluid equals the weight of the fluid displaced by the object. Mathematically, the buoyant force \(F_b\) is \(F_b = \rho_{\text{outside}} \cdot V \cdot g\), where \(\rho_{\text{outside}}\) is the density of the cooler outside air, \(V\) is the volume of displaced air, and \(g\) is acceleration due to gravity.
4. Compare the weight of the balloon (including the heated air inside) to the buoyant force. Since the heated air inside has lower density, the total weight of the balloon plus hot air is less than the weight of the displaced cooler air outside, resulting in a net upward force.
5. Conclude that this net upward buoyant force causes the hot-air balloon to rise in the atmosphere.