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The increased pressure within the ball (to a point) brings about increased elasticity in the gas within the ball. Elasticity is the degree to which an object is able to release the energy it receives through a collision, without losing energy to entropy of any kind. No object is perfectly elastic. The energy of the kick will produce energy that goes into deforming the ball and adding heat to it. The gas inside the ball with compress and expand, and there will be some heat in that process as well. These are forms of energy that do not add to the ball's recoil. But the more pressure inside the ball, the more efficient the expansion of the gas becomes, after the kick's compression. Imagine kicking a completely deflated ball. There is little or no elasticity, and the ball will absorb a lot of the kick's energy just in the process of being deformed.

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Q: Why does a football with more air pressure go furthur when you punt it than a football with less air pressure?
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