00:00 The physics of hot air balloons: In this hot air balloon problem, we use physics to learn how a hot air balloon works. We are given the mass of the balloon material, equipment and passengers, and we want to find the volume of hot air required to lift a balloon. Given the density of the cold outside air and given the density of the hot air inside the balloon, we plan to use Archimedes' principle to compute the buoyant force on a hot air balloon then calculate the minimum volume of hot air required to balance the force of gravity. We work the problem twice: once using the approximation that the buoyant force on the passengers and equipment is negligible, then we relax this assumption to make a small correction to the result for the balloon volume.
00:53 A couple important points before we start the physics: first, the mass of the hot air inside the balloon cannot be ignored. It actually turns out that the vast majority of the mass of the balloon system belongs to the hot air! Second, we get a short reminder of the definition of density rho=mass/volume, which means we can quickly compute masses by writing mass=density*volume. These calculations happen repeatedly in Archimedes' principle problems.
01:28 Balance forces to start out the solution: we have gravity pulling down on the balloon system and buoyant force pointing up, and to get the balloon to float we use the minimum volume of hot air to balance the vertical forces on the balloon. The buoyant force is approximated by pretending the balloon has all the volume (ignoring the buoyant force on passengers and equipment for this first approximation). So the buoyant force is equal to the weight of displaced cold air, or density of cold air*Volume of the balloon*g. The weight comes from two parts: the mass of the equipment and passengers and the weight of the hot air in the balloon. The weight of the hot air is density of hot air*Volume*g. With the forces balanced, we can solve for the volume of the balloon.
05:05 The mass of the hot air is not negligible! Now that we have the volume of the hot air balloon, we can actually compute the mass of the hot air by using density of hot air*Volume*g, and it turns out that the hot air accounts for the vast majority of the mass of the balloon system!
05:40 Relax the assumption that the buoyant force on the passengers and equipment is negligible: we improve on our first answer by repeating the calculation, but this time we include the buoyant force on the passengers and equipment. We are given the density of these parts and we know the mass, so we use the density formula solved for volume=mass/density to get the volume of the passengers and equipment. This allows us to use Archimedes' principle to get the buoyant force on the passengers and equipment, and we put this small correction into the force equation. Solving for V once again, we find a small but noticeable correction to the volume of hot air required to lift the balloon.