Newton's laws of motion | Third law

Newton’s Third Law and Conservation of Momentum

Newton’s laws of motion form the foundation of classical mechanics, and two of their most profound applications are action-reaction pairs (Newton’s third law) and the conservation of linear momentum. These principles govern interactions in everyday phenomena—from sports collisions to rocket propulsion. This article breaks down these concepts with real-world examples, mathematical insights, and common misconceptions.

1. Action-Reaction Pairs: The Core of Newton’s Third Law

Newton’s third law states:

“For every action, there is an equal and opposite reaction.”

This means forces always occur in pairs:

• Action: A force exerted by Object A on Object B.
• Reaction: A force of equal magnitude but opposite direction exerted by Object B on Object A.

Key Characteristics:

• Equal Magnitude: The forces are always equal in strength.
• Opposite Direction: The forces act in opposite directions.
• Different Effects: Effects differ based on the objects' masses (accelerations may differ).

Real-World Examples:

• Swimmer Pushing Water: Push water backward (action), water pushes swimmer forward (reaction).
• Rocket Propulsion: As the exhaust gases are pushed downward (action), the rocket propels upward (reaction) according to Newton’s third law.

2. Conservation of Linear Momentum: The Unchanging Total

The principle of linear momentum conservation explains that:

“In an isolated system (no external forces), the total momentum before a collision equals the total momentum after the collision.”

Mathematically:

\(\overrightarrow{P_{initial}}=\overrightarrow{P_{final}}\)

Where momentum is defined as:

\(\overrightarrow{p}=\overrightarrow{mv}\)

Derivation from Newton’s Third Law:

Consider two colliding objects with masses \( m_1 \) and \( m_2 \).

During collision:

\(\overrightarrow{F_{12}} = -\overrightarrow{F_{21}}\)

Impulse \(\overrightarrow{F}\triangle t\) causes equal and opposite momentum changes:

\(m_1 \triangle\overrightarrow{v_1} = - m_2 \triangle\overrightarrow{v_2}\)

Rearranged, this gives:

\(m_1 \overrightarrow{v_1} + m_2 \overrightarrow{v_2}= constant\)

Applications of Momentum Conservation:

• Car Crashes: Heavier vehicles undergo smaller velocity change.
• Sports: After colliding, the football players move as a single unit.
• Gun Recoil: When a firearm is discharged, the bullet accelerates forward due to explosive force. Simultaneously, the gun undergoes an equal and opposite recoil, exemplifying the conservation of momentum where the forward motion of the bullet is counterbalanced by the gun’s backward motion.

3. Connecting Action-Reaction Pairs to Momentum Conservation

Action-reaction forces explain how total momentum is conserved. During internal interactions (like collisions), these forces cancel each other, keeping the system’s total momentum unchanged.

Example: When a moving ball hits a stationary ball:

• The first ball loses momentum (action).
• The second ball gains equal momentum (reaction).

4. SEO-Optimized FAQ

Q1: Why don’t action-reaction forces cancel each other?

Each force is applied to a different object. For example, when you push a wall, the wall pushes back, but these forces act on separate entities.

Q2: How can a rocket travel in space where there is no air to push off of?

The rocket pushes exhaust gases downward, causing an upward force on the rocket in return, following Newton's third law.

Q3: Why does a gun recoil when fired?

The forward momentum of the bullet is exactly balanced by the backward recoil momentum of the gun, keeping the system’s total momentum conserved.