Final answer:
This question addresses the physics of a basketball's bounce, related to energy losses in elastic collisions, and the change in a volleyball's density with increased volume. The correct inflation of a basketball affects its bounce height, as seen in pre-game checks by officials, and a volleyball's density decreases if the volume increases without adding mass.
Step-by-step explanation:
The question involves the concept of an exponential decrease in the height of a basketball bounce, which is related to elastic collisions and energy conversion in physics. When a basketball is inflated to the correct pressure, it bounces back to a certain height after being dropped; this height decreases with each bounce due to energy losses primarily in the form of sound and heat. For example, a ball dropped from a height of 1 m might bounce back to 0.8 m, then 0.5 m, and further down to 0.2 m confirming the exponential nature of the decline. The phenomenon is a result of kinetic energy not being fully conserved during the collision with the ground; some of the energy is converted to other forms, which is why the ball doesn't return to its original height. Officials before a game ensure the ball is inflated properly by checking if it bounces to a specific height; if not, they adjust the pressure accordingly. In another context, understanding the mechanics involved can help determine the required initial velocity for a basketball player to achieve a certain jump height, like rising 1.25 m above the floor.
Regarding the question about volleyball density, an increase in the radius of the volleyball by 10% without adding mass will result in a decrease in density. Given the volume of a sphere is proportional to the cube of the radius, the volume increases by a factor larger than the percentage increase in radius, causing its density to drop. This is because density is mass over volume, and when only volume increases, density decreases.