Foundations — Newton's third law — action-reaction, common misconceptions

Before you can use Newton's third law you must be able to read it. The parent note throws around arrows, subscripts, masses, momenta and derivatives as if you already knew them. Here we build each one from absolute zero, in an order where every symbol is earned before it is used.

---

1. What is an "object" or "body"?

The picture is the whole point: the moment we choose what the body is, we draw an imaginary bubble around it. Anything inside the bubble is "the body"; everything outside is "the rest of the universe" that can push or pull on it.

Why the topic needs this. The third law's punchline is "the two forces act on two different bodies." That sentence is meaningless until you can point at exactly which bubble each force lives in. Most misconceptions in the parent note come from putting two forces in the same bubble by accident.

---

2. Arrows: what a vector is

We write a vector with a little arrow on top, like $\vec{F}$. The plain letter without the arrow, $F$, means only its size (also called its magnitude), never its direction.

The tool "put a minus sign in front of a vector," $-\vec{F}$, means: keep the same length, flip the arrow to point exactly backward. This is the single most important notation on the whole page — the third law is the sentence ${\vec{F}}_{A\to B}=-{\vec{F}}_{B\to A}$, which reads "same length, flipped direction."

---

3. Force: the push or pull

A force is always one body acting on another body. There is no such thing as a force that just floats around belonging to nobody. This is why we need labels.

Read it aloud every time until it is automatic:

• ${\vec{F}}_{A\to B}$ ::: "force from A on B."
• ${\vec{F}}_{B\to A}$ ::: "force from B on A."

The third law says these two — with the labels swapped — are equal-and-opposite. Swapping $A\leftrightarrow B$ is literally the mnemonic's first "S."

Why the topic needs this. Rules 4 and 5 of the parent's "true pair" checklist are entirely about these subscripts. Without the $A\to B$ notation you cannot even state whether two forces are a genuine pair.

---

4. Types of force (why "same type" matters)

The parent's rule 3 says a true pair must be the same type of force. So we need names for the common types:

Symbol	Plain meaning	Picture
gravity ${\vec{F}}_{E\to b}$	the Earth pulling a body down	arrow straight down toward Earth's centre
normal $\vec{N}$	a surface pushing outward, perpendicular to itself	arrow at 90° out of the table/floor
friction $\vec{f}$	a surface dragging along itself, opposing sliding	arrow flat along the surface

You meet these in detail in Normal Force and Friction. For now the only thing that matters: normal and gravity are different types, so they can never be each other's partner — that kills the parent's third misconception in one line.

---

5. Mass $m$: how hard to shove

The picture: a bowling ball and a beach ball may be the same size, but the bowling ball is much harder to kick into motion. That "harder to get moving" is mass.

Why the topic needs this. The parent's biggest "aha" — a truck and a mosquito feel equal forces but have wildly different fates — is entirely about mass. Same force $F$, but $a=F/m$: the tiny-mass mosquito gets a monster acceleration.

---

6. Velocity $\vec{v}$ and momentum $\vec{p}$

Because $\vec{p}$ carries a mass factor, a light body and a heavy body can have the same momentum at very different speeds — which is exactly why the two ice skaters in the parent (60 kg vs. 40 kg) fly apart at different speeds while sharing one momentum-swap. You will use this heavily in Conservation of Momentum.

---

7. The rate-of-change symbol $\frac{d}{dt}$

The parent's derivation writes $\frac{d\vec{P}}{dt}$. This looks scary; it is not.

Two facts are all you need:

1. A constant has zero rate of change: if $\vec{P}$ never changes, $\frac{d\vec{P}}{dt}=0$.
2. Rate of change of momentum = force. This is Newton's *second* law in its truest form: $\frac{d\vec{p}}{dt}=\vec{F}$. Force is nothing but how fast momentum changes.

Now the parent's whole derivation reads in plain words: total motion of an isolated pair never changes (rate = 0); rate of change of each body's motion is the force on it; so the two forces must add to zero — i.e. they are equal and opposite. That is the third law, born from momentum.

---

8. The free-body diagram (the workhorse tool)

The golden rule that dissolves the parent's #1 misconception ("action and reaction cancel"): a 3rd-law pair never appears in the same FBD, because the two forces act on two different bodies, which get two different diagrams. Forces can only cancel within a single FBD. You will drill this in Free Body Diagrams.

---

How the foundations feed the topic

Every arrow above is a "you cannot understand the box it points to until you understand the box it comes from." The whole page exists to fill in the left-hand boxes.

---

Equipment checklist

Cover the right side and test yourself. If any answer is fuzzy, reread that section before touching the parent note.

• What does the little arrow on $\vec{F}$ mean, and what does plain $F$ mean? ::: $\vec{F}$ has size and direction (an arrow); plain $F$ is only the size (magnitude).
• In ${\vec{F}}_{A\to B}$, which body actually feels the force? ::: Body $B$ — the arrow in the subscript points to the one being pushed.
• What does the minus sign do in $-\vec{F}$? ::: Keeps the same length, flips the direction exactly backward.
• What is mass, in one phrase? ::: How stubborn a body is — its resistance to being sped up or slowed.
• Write momentum as a formula and say what it pictures. ::: $\vec{p}=m\vec{v}$; the body's "quantity of motion" / unstoppability.
• What does $\frac{d\vec{P}}{dt}=0$ say in plain words? ::: The total momentum is not changing in time — it is constant.
• Which law says $\frac{d\vec{p}}{dt}=\vec{F}$? ::: Newton's second law — force is the rate of change of momentum.
• Why can a 3rd-law pair never cancel? ::: The two forces act on two different bodies, so they live in two different free-body diagrams; cancellation needs forces on the same body.
• Are normal force and gravity the same type of force? ::: No — one is a contact push perpendicular to a surface, the other is Earth's pull; different types can never be partners.
• Same force on a truck and a mosquito — why different outcomes? ::: $a=F/m$; the mosquito's tiny mass gives a huge acceleration.

---

Connections

• Newton's Second Law — supplies $\vec{F}=m\vec{a}$ and the momentum form $d\vec{p}/dt=\vec{F}$ used here.
• Conservation of Momentum — the fact "$d\vec{P}/dt=0$" that the whole derivation stands on.
• Free Body Diagrams — the tool that keeps each force in the correct bubble.
• Normal Force and Friction — the force types whose distinctness kills the "normal = reaction to gravity" myth.
• Rocket Propulsion & Variable Mass — where momentum and the third law combine.
• Center of Mass Motion — where "total momentum of a system" gets its natural home.