Increasing a ball’s air pressure increases its bounce. The more air there is inside a ball, the more it resists deformation when it meets a rigid object, enabling it to bounce more elastically.
This relation between air pressure and bounce is critical in the world of professional sports. U.S. Soccer regulations contain a passage in Law 2 mentioning that soccer balls used in professional soccer matches must be inflated to between 8.5 and 15.6 psi at sea level.
A ball’s bounce is a complex function of its material and internal pressure as well as the ambient pressure and temperature outside the ball. The more elastic the ball material is, the more it bounces when it is properly inflated. At a molecular level, pressure results when the molecules of the air inside the ball randomly collide with each other and with the inside of the ball, pushing out against the ball material. The higher the pressure inside the ball and the more molecules there are to push against the ball interior, the stiffer the ball becomes.
A ball colliding with a rigid surface deforms, increasing the air pressure around the deformed region. This additional pressure spreads through the ball interior, causing the ball to bounce off the ground.