Animated Solution for Physics - System of Particles: When a ball is thrown up, the magnitude of its momentum decreases and then increases. Does this violate the conservation of momentum principle?

Imagine standing in an open field and tossing a ball straight up into the air. As it leaves your hand, it has a certain velocity, and therefore, a certain momentum. As it climbs higher, gravity pulls it down, slowing it until it momentarily stops at its peak. At this exact instant, its momentum is zero. Then, it begins its descent, speeding up, and its momentum increases once again.

At first glance, this seems like a blatant violation of one of the most sacred laws of physics: the Conservation of Linear Momentum. If momentum is supposed to be conserved, how can the ball's momentum just disappear and reappear?

To resolve this paradox, we must look at the fine print of the conservation law. The principle states that the total linear momentum of a system remains constant only if the net external force acting on that system is zero.

When we look at the ball in isolation, we are defining our "system" as just the ball. Is there an external force acting on this system? Yes! The Earth is exerting a gravitational pull downwards on the ball. Because there is a net external force ($F_{\text{ext}}=mg$), the momentum of the ball alone is not required to be conserved. It is perfectly legal for its momentum to change.

The Ball-Earth System

Now, let's zoom out. What if we redefine our system to include both the Ball and the Earth?

In this new, expanded system, the gravitational pull of the Earth on the ball is no longer an external force. It is an internal force between two members of the same system. By Newton's Third Law, the ball also exerts an equal and opposite gravitational pull on the Earth. These internal forces cancel each other out perfectly.

Assuming no other celestial bodies are interfering, the net external force on the Ball-Earth system is exactly zero. Therefore, the total momentum of the Ball-Earth system must be conserved!

If the total momentum is conserved, what happens when the ball is moving upwards with momentum $m\vec{v}$? To keep the total momentum of the system constant (let's say it was initially zero before you threw it), the Earth must acquire an equal and opposite momentum.

This means the Earth actually recoils downwards with a velocity $\vec{V}$!

Because the mass of the Earth ($M$) is astronomically larger than the mass of the ball ($m$), the Earth's recoil velocity $\vec{V}$ is infinitesimally small—so small that it is completely undetectable. But mathematically and physically, it is there. As the ball slows down, the Earth's recoil slows down. When the ball stops at the top, the Earth stops. As the ball falls back down, the Earth "falls" up to meet it.

So, does the changing momentum of the ball violate the conservation of momentum? Absolutely not. It simply reminds us that we must always be careful about how we define our system.