Animated Solution for Physics - System of Particles: A ball hits the floor and rebounds after an inelastic collision. In this case,

Have you ever dropped a tennis ball and watched it bounce? It seems like such a simple, everyday occurrence. But beneath that simple bounce lies a profound symphony of physics—a delicate interplay of energy, momentum, and the fundamental laws that govern our universe.

In this problem, we are asked to analyze a ball hitting the floor and rebounding after an inelastic collision. We need to determine which physical quantities are conserved and which are lost. Let's break this down step-by-step and uncover the hidden reality of the bounce.

When you drop a ball, it accelerates downwards due to gravity, gaining kinetic energy. Right before it hits the floor, it has a maximum downward velocity, which we can call ${\vec{v}}_i$.

Then comes the impact. The ball squishes against the floor, comes to a momentary halt, and then springs back up. But notice the wording in the problem: it is an inelastic collision.

In the real world, almost all macroscopic collisions are inelastic. When the ball deforms, the molecules inside it rub against each other, generating friction. This friction converts some of the ball's precious kinetic energy into thermal energy (heat) and acoustic energy (the "thud" sound you hear).

Because this energy is lost to the environment, the ball rebounds with a smaller velocity, ${\vec{v}}_f$, and fails to reach its original drop height.

Let's look at the options provided in the question. Option (a) suggests that the momentum of the ball is conserved. Is this true?

Momentum is a vector quantity, defined as $\vec{p}=m\vec{v}$. It has both magnitude and direction. Before the collision, the ball's momentum is pointing straight down. After the collision, it is pointing straight up. Even if the ball bounced back with the exact same speed (which it doesn't), the direction of the momentum has completely reversed. Therefore, the momentum of the ball itself is absolutely not conserved.

What about Option (b), which claims the mechanical energy of the ball remains the same? Mechanical energy is the sum of kinetic and potential energy. As we just discussed, the inelastic nature of the collision means kinetic energy is permanently lost as heat and sound. Consequently, the mechanical energy of the ball decreases. Option (b) is incorrect.

The Ball-Earth System

To find what is truly conserved, we need to change our perspective. Instead of just looking at the ball, let's zoom out and consider the ball and the Earth as one giant, combined system.

When the ball hits the floor, the floor pushes up on the ball (the normal force), and the ball pushes down on the floor with an equal and opposite force (Newton's Third Law). If our system is just the ball, the floor's push is an external force, which changes the ball's momentum.

But if our system includes both the ball and the Earth, these forces are entirely internal. They are just different parts of the system pushing on each other.

What about gravity? Gravity is the Earth pulling on the ball, and the ball pulling on the Earth. Again, if the Earth is part of our system, gravity is an internal force!

Newton's Second Law states that the rate of change of momentum of a system is equal to the net external force acting on it:

Since there are no giant, cosmic hands pushing on our Ball-Earth system from the outside, the net external force is zero (${\vec{F}}_{\text{ext}}=0$).

Therefore, the total linear momentum of the system cannot change. Whatever momentum the ball loses (or gains in the opposite direction), the Earth must acquire in equal measure. Because the Earth is so incredibly massive, its resulting change in velocity is infinitesimally small—so small we can't measure it—but it is mathematically real.

Thus, the total momentum of the ball and the Earth is perfectly conserved. This makes Option (c) the correct answer.

Finally, let's quickly dismiss Option (d), which suggests the total mechanical energy of the ball and Earth is conserved.

Remember that heat and sound were generated during the inelastic impact. These are non-mechanical forms of energy. While the total energy of the universe is always conserved (First Law of Thermodynamics), the mechanical portion of that energy is not. The energy dissipated into the floor and the air is lost from the mechanical ledger forever.

This problem is a beautiful reminder of why defining your system is the most crucial step in physics. By simply expanding our boundary to include the Earth, a messy, energy-losing collision reveals a perfectly conserved quantity. The next time you bounce a ball, remember: you aren't just moving the ball; you are moving the entire planet, just a little bit, to keep the universe in balance.