Momentum Conservation from Newton's Third Law

1. From Newton's Third Law to Momentum Conservation

Let's imagine a simple interaction: a cannon firing a cannonball. The cannon pushes the ball forward, and the ball pushes the cannon backward. These two pushes are the "action-reaction" pair described by Newton's Third Law. The forces are equal in magnitude and opposite in direction. But what does this mean for their momentum?

To see the connection clearly, let's watch a video that derives the conservation of momentum step-by-step from this very idea.

The video's derivation is the core of today's lesson. Let's formalize the key steps:

1. Newton's Third Law: When two objects (A and B) interact, the force object A exerts on B is equal and opposite to the force object B exerts on A.
   ${\vec{F}}_{A\text{ on }B}=-{\vec{F}}_{B\text{ on }A}$

2. Impulse: The interaction occurs over the same time interval, $\Delta t$, for both objects. If we multiply both sides of the equation by $\Delta t$, we get the impulse for each object.
   ${\vec{F}}_{A\text{ on }B}\Delta t=-{\vec{F}}_{B\text{ on }A}\Delta t$
   This means the impulse on object B is equal in magnitude and opposite in direction to the impulse on object A.
   ${\vec{J}}_B=-{\vec{J}}_A$

3. Impulse-Momentum Theorem: From our last lesson, we know that impulse equals the change in momentum ($\vec{J}=\Delta \vec{p}$). Substituting this in gives us the final, crucial relationship:
   $\Delta {\vec{p}}_B=-\Delta {\vec{p}}_A$

This equation is powerful. It says that during any interaction, the momentum gained by one object is exactly equal to the momentum lost by the other object. Momentum is transferred, but none is lost.

2. The Principle of Conservation of Momentum

The relationship $\Delta {\vec{p}}_B=-\Delta {\vec{p}}_A$ can be rearranged by moving both terms to one side:
$\Delta {\vec{p}}_A+\Delta {\vec{p}}_B=0$

This tells us that the total change in momentum for the system of two objects is zero. If the total momentum doesn't change, it must be conserved. This leads us to the formal statement of the law of conservation of momentum.

As the article explains, if the total momentum doesn't change, then the total momentum before the collision must equal the total momentum after the collision. For a system of two objects, this is written as:

${\vec{p}}_{A,\text{initial}}+{\vec{p}}_{B,\text{initial}}={\vec{p}}_{A,\text{final}}+{\vec{p}}_{B,\text{final}}$

Or, more generally:
$\sum {\vec{p}}_{\text{initial}}=\sum {\vec{p}}_{\text{final}}$

This principle is one of the pillars of physics. It holds true from subatomic particles to colliding galaxies.

3. Isolated Systems: The Fine Print

There is one critical condition for the law of conservation of momentum to hold: the system must be isolated (or closed). This means there are no net external forces acting on the system.

• Internal Forces: The forces that the colliding objects exert on each other. As we saw, these always come in equal and opposite pairs and cancel out, so they cannot change the system's total momentum.
• External Forces: Forces exerted by things outside the system (like gravity, friction, or air resistance). If these forces don't cancel each other out, they will change the system's total momentum.

Let's return to the Khan Academy video to explore this crucial distinction.

To summarize the key idea: for a system to be considered isolated, the net external force must be zero (${\vec{F}}_{net,ext}=0$).

• On a frictionless horizontal surface, gravity (down) and the normal force (up) are external forces, but they cancel out. So, the system is isolated in the horizontal direction, and momentum is conserved horizontally.
• For very short, high-force events like collisions and explosions, the internal forces are so massive compared to external forces like friction that we can approximate the system as isolated for that brief moment.

4. Check Your Understanding

Let's test this knowledge with a classic physics puzzle.

Consider a head-on collision between a large truck and a small Volkswagen car.

1. Which vehicle experiences the greater force of impact?
2. Which vehicle experiences the greater impulse?
3. Which vehicle experiences the greater change in momentum?
4. Which vehicle experiences the greater acceleration?

Think about this using Newton's Third Law and the Impulse-Momentum theorem.

Conclusion

Today we connected two major concepts in physics. We saw that the familiar idea of action-reaction pairs is the fundamental reason why momentum is conserved in interactions.

Key Takeaways:

• The principle of conservation of linear momentum is a direct consequence of Newton's Third Law.
• In any interaction, the momentum change of one object is equal in magnitude and opposite in direction to the momentum change of the other object ($\Delta {\vec{p}}_A=-\Delta {\vec{p}}_B$).
• Because of this, the total momentum of an isolated system (one with no net external force) remains constant before, during, and after an interaction.
• This principle is extremely useful for analyzing collisions and explosions, especially when the interaction happens over a very short time.

Preview of the Next Lesson:
We've established that the total momentum of an isolated system is always conserved in a collision. But what about kinetic energy? Is it also conserved? As we'll see next time, the answer is "sometimes." This will lead us to distinguish between two types of collisions: elastic and inelastic. This distinction is key to solving a wider range of collision problems, including those you might encounter in robotics.