What air force makes a air balloon go up and down?

Answer 1

This is the buoyancy force. Archimedes is credited with first formulating this into a mathematical principle: the buoyancy force of an object is equal to the weight of fluid it displaces. In the case of a hot air balloon, the "fluid" is the outside cool air and the balloon is displacing a volume of cool air equal to the volume of the inflated balloon. So you can say the buoyancy force F = V ρ g, where V is the volume of the inflated ballon, ρ is the outside air density, and g is gravity.

A balloon can float because it is displacing this cool air with hot air which is less dense. The volume of hot air inside the balloon thus weighs less than the same volume of cold air outside the balloon. For a balloon to be float upwards, the buoyancy force has to be at least equal to the weight of the balloon (the balloon fabric, the gondola, the people, equipment and cargo, and the hot air inside).

For example, for a balloon of 100,000 cubic foot volume (typical), with outside air at 20 deg C near sea level, the buoyancy force is about 7500 lbs. This force has to lift the gondola, people, equipment and of course the hot air inside the balloon, all of which have weight. By far the greatest percentage of that total weight is the heated air inside since there is so much of it.