Explain Newton's third law with an example

Newton's Third Law

Newton's third law states:
For every action, there is an equal and opposite reaction.

In other words, if object A exerts a force $\vec{F}_{A on B}$ on object B, then object B exerts an equal force $\vec{F}_{B on A} = -\vec{F}_{A on B}$ on object A.

The forces are equal in magnitude, opposite in direction, and act on different objects.

Step 1: State the law mathematically
$\vec{F}_{A on B} = -\vec{F}_{B on A}$

Step 2: Example — Person pushing against a wall
Consider a person pushing on a wall with their hand.

Action force: Hand exerts force $\vec{F}_{hand on wall}$ to the right on the wall (magnitude $F$, say 50 N).
$\vec{F}_{hand on wall} = F\hat{i} \quad (50~N to the right)$

Reaction force: Wall exerts equal and opposite force on hand.
$\vec{F}_{wall on hand} = -F\hat{i} \quad (50~N to the left)$

Step 3: Verify equality
Magnitude: $|\vec{F}_{hand on wall}| = |\vec{F}_{wall on hand}| = 50~\text{N}$
Direction: Opposite (right vs. left)
Objects: Different (hand vs. wall)

The person feels the push back from the wall, but neither moves much due to the wall's large mass.

Step 4: Real-world implication
These forces don't cancel because they act on different objects. The net force on each determines acceleration via Newton's second law: $\vec{F}_{net} = m\vec{a}$.