A ball filled with air is basically a spring. Pushing the air molecules closer together (i.e., fully inflating the ball) makes the "spring" stiffer. When the ball is about to hit the ground, it has some energy (capacity to do something). Because the ball is moving, this energy is called kinetic energy. As the ball hits the ground, the kinetic energy is stored as potential energy in the "spring" as the ball deforms and the air inside is compressed. But some of the kinetic energy is lost to friction (becoming heat) as the skin of the ball is deformed. Thus, less energy will be available to push the ball back up and it will not bounce as high. In a partially-inflated ball, the air molecules are not very close together, so pushing them nearer each other (when the ball hits the ground) is relatively easy. In a fully-inflated ball the air molecules are already packed close and further reducing that separation is difficult. Thus, the less inflated ball will deform more than the fully inflated ball, in the process losing more useful energy to friction, and then having less energy to bring the ball back up.

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Very good question! The answer lies in Isaac Newton’s third law. The law states that for every action, there is an equal and opposite reaction. That means when you sit in your seat, the seat pushes back to hold you up so you don’t fall right through the chair. Now let’s consider a ball, like a basketball. The inside of the ball only has so much space, so the more air you pump into it, the more squeezed together the air gets. This is called air pressure. Now the air inside is pushing so hard on the skin of the ball that when it hits the ground, it hits the ground really hard! And remember Newton’s third law? If the ball hits the ground really hard, the ground is going to hit the ball back just as hard. That’s why the less inflated ball won’t bounce as high. The air particles inside aren’t as squished and are just fine with hanging out inside the ball with minimal effort to push on the walls.

It's because the air inside of the inflated ball is under pressure, while the air inside of the deflated ball is under less pressure. Pressure means force, and Newton's third law says that when you push on something, that something pushes back. You can give an example of pushing on a wall: if you push on a wall, the wall pushes back to keep you from going through it. If you push on the wall hard enough, you will push yourself away from the wall. The inflated ball when it hits the ground it pushes so hard that the ground pushes the ball back up into the air - in other words, it bounces.

What a fun challenge! Explaining this physics to second graders!

The less inflated ball is sort of squishy, so when it hits the floor, some of the energy goes into squishing the ball, and there's not so much energy left for making the ball bounce back up. The fully inflated ball is stiffer or harder, so when it hits the floor, pretty much all of the energy goes into making the ball bounce back up.

Newton's Cradle could show this - stick things to a ball in the 'cradle' and see how it affects the bouncing. Tape vs clay etc. They things would damp the balls' bouncing by various amounts. But 2nd graders could quickly wreak havoc on an unattended 'cradle.' A young neighbor kid messed with my last one till it was tangled. Luckily it was a tiny cheap 'cradle' so I didn't care.