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The key concept here is density. Less dense objects will rise to the top of a more dense medium while more dense objects will sink to the bottom of a less dense medium. In the case of a hot air balloon floating in the sky, we are talking about hot air versus cold air here. The hot air in the balloon is less dense that the cold air medium that surrounds it, so the hot air in the balloon will lift it higher in the air. In the case of a boat on water, we are talking about air versus water. Since air is much less dense than water, the air trapped behind the hull of the boat will keep the entire boat floating on top of the water. In terms of forces, the more dense medium (cold air or water) exerts an upward force against the less dense object (hot air). If you drop a penny in water, however, it will instantly sink since copper (or metal in general) is much more dense than water.
A boat floating in water, or a hot air balloon suspended in the air are examples of buoyancy. It was Archimedes who discovered the law of buoyancy, namely that a body immersed or suspended in a liquid or gas is buoyed up by a force equal to the weight of the liquid or gas displaced by the object.
In the case of a boat, the force of buoyancy is equal to the weight of water displaced by the boat. As the boat enters the water it will sink into the water until enough water is displaced to equal the weight of the boat and its contents. Thus, as weight is added to a given boat, the water line will rise.
A hot air balloon rises because hot air is less dense than cold air. This is because the molecules of air are driven farther apart from one another as the temperature increases. The air becomes less dense.
The balloon displaces its entire volume. So the volume of the balloon is equal to the volume of the air it displaces. By Archimedes Law, the balloon is buoyed up by a force equal to the weight of the displaced (cold) air, which is greater than the weight of an equal volume of the lighter hot air contained in the balloon. Hence, the balloon rises.
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