Newton's Laws of Motion¶
The Formulas¶
First Law (Inertia):
Second Law (Force):
Third Law (Action-Reaction):
What They Mean¶
These three laws are the entire foundation of classical mechanics. Every bridge, every rocket trajectory, every car crash simulation — all of it comes down to these three statements.
The first law says: nothing changes its motion unless something forces it to. The second law says: how much it changes depends on how hard you push and how heavy it is. The third law says: you can't push without being pushed back.
Together, they replaced 2,000 years of Aristotelian physics with something that actually works.
Why They Work — The Story Behind the Laws¶
The 2,000-Year Mistake¶
For two millennia, everyone believed Aristotle. He taught that the "natural state" of an object is rest. A thrown stone slows down because it "wants" to stop. A cart needs a horse pulling it because without force, motion ceases.
This sounds perfectly intuitive. In everyday life, things do stop moving. Push a box across the floor and let go — it stops. Aristotle's physics matched what people saw every day.
But there was a crack in the logic. If a thrown stone is propelled by the air closing behind it (Aristotle's explanation), why doesn't a stone with a flat back fly farther than a round one? Why does an arrow with feathers fly better than one without? Nobody had good answers.
Galileo's Ramps: The Breakthrough (1630s)¶
Galileo Galilei didn't have fancy equipment. He had polished ramps and a water clock. But he had something more important: the willingness to question what "everyone knew."
He rolled brass balls down an inclined ramp and up another on the other side:
- The ball rolled down, gained speed, and climbed the second ramp to almost the same height
- He made the second ramp less steep — the ball rolled farther but still reached the same height
- Then Galileo asked the killer question: "What if the second ramp is perfectly flat?"
The ball would never reach the original height. It would just keep rolling. Forever.
This was revolutionary. Galileo realized that friction was what stopped things, not some innate desire for rest. Without friction, motion would continue indefinitely. "Rest" isn't special — it's just what happens when something gets in the way.
Aristotle had been fooled by friction for 2,000 years.
Newton's Synthesis: The Principia (1687)¶
Isaac Newton was 23 years old, stuck at home during the Great Plague of 1665-66, when the ideas started forming. It took him another 20 years to publish them.
In 1687, he released Philosophiæ Naturalis Principia Mathematica — arguably the most important scientific book ever written. In it, he took Galileo's insight and built a complete mathematical framework that could predict the motion of everything from cannonballs to comets.
The Three Laws, Derived¶
First Law: Inertia¶
"Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon."
This is Galileo's ramp experiment stated as a universal principle. Newton is saying: the universe has no preference. Moving is just as natural as sitting still. Force doesn't cause motion — force causes change in motion.
Why is this a "law" and not just common sense? Because it defines what a force is. Before Newton, people thought force was what kept things moving. Newton flipped it: force is what changes motion. No force, no change. This subtle shift is the foundation of everything that follows.
Second Law: \(F = ma\)¶
This is the big one. And it raises a question that most textbooks skip: how did Newton arrive at this? Did he run experiments? Was it a guess? A definition?
The answer is: it was a brilliant chain of reasoning, built on top of Galileo's experimental work.
Step 1: Galileo's Data¶
Galileo had already established something crucial with his ramp experiments: when gravity pulls an object, it doesn't give it a constant speed — it gives it a constant acceleration. The ball on the ramp doesn't instantly reach full speed. It speeds up steadily, gaining the same amount of velocity every second.
He measured this precisely: distance covered grows as the square of time (\(d \propto t^2\)). This is the hallmark of constant acceleration. An object falling for 2 seconds covers 4 times the distance of one falling for 1 second. Not 2 times — 4 times.
Step 2: Newton's Insight — What "Force" Really Does¶
Newton looked at Galileo's result and asked: what does it mean that gravity produces constant acceleration?
It means that the effect of a force isn't to produce motion. If it were, a constant force would produce constant speed. Instead, a constant force produces constant change in speed — acceleration. Force doesn't move things. Force accelerates things.
This was the conceptual leap. Newton redefined force entirely. For Aristotle, force was what kept a cart moving. For Newton, force was what changed how the cart moved. No force = no change = constant velocity (including zero).
Step 3: The Role of Mass¶
Newton also needed to account for something obvious: the same push has different effects on different objects. Kick a football and it flies. Kick a boulder and your foot breaks.
He defined "quantity of motion" (what we now call momentum) as:
Why \(mv\)? Because experiments on collisions showed that this quantity was conserved. When two billiard balls collide, the total \(mv\) before equals the total \(mv\) after. This wasn't a guess — it was observed by Huygens, Wren, and Wallis in careful experiments reported to the Royal Society in the 1660s.
Step 4: Putting It Together¶
Newton's second law states that force is the rate at which momentum changes:
If the mass stays constant (which it does for most everyday situations), this simplifies:
Is this a definition or a discovery? It's both — and that's what makes it so clever. Newton defined force as what causes acceleration. But the empirical content is that this definition works consistently. The same force always produces the same acceleration on the same mass, regardless of when, where, or how you apply it. That's not guaranteed by the definition — that's a fact about the universe.
Step 5: The Ultimate Test — Predicting the Cosmos¶
The proof that \(F = ma\) wasn't just a clever definition came when Newton combined it with his law of gravitation (\(F = GMm/r^2\)) and showed that it predicted Kepler's three laws of planetary motion — laws that had been painstakingly extracted from decades of astronomical observation by Tycho Brahe and Johannes Kepler.
Nobody told Newton what the planets should do. He derived their orbits from \(F = ma\) and gravity, and the result matched reality. That's how you know it's not just a definition. It's a law of nature.
The deeper version (\(F = dp/dt\)) is more general. It handles cases where mass changes too — like a rocket burning fuel, where both \(m\) and \(v\) change simultaneously. Newton was clever enough to write the law in the more general form from the start.
Third Law: Action and Reaction¶
"To every action there is always opposed an equal reaction."
This was the missing piece for understanding the cosmos. If the Earth pulls the Moon, does the Moon pull the Earth? Aristotle would say no — the Earth is the center of everything, it doesn't get pulled around.
Newton said yes. Every force comes in pairs. You cannot push something without it pushing you back. When you stand on the floor, you push the floor down with your weight, and the floor pushes you up with exactly the same force. That's why you don't fall through it.
The key insight: forces are interactions between two objects, not properties of a single object. There's no such thing as an isolated force. This allowed Newton to prove that the same gravity pulling an apple down was also pulling the Moon toward the Earth — and the Earth toward the Moon.
Variables Explained¶
| Symbol | Name | Unit | Description |
|---|---|---|---|
| \(\vec{F}\) | Force | Newtons (N) | A push or pull that changes motion |
| \(m\) | Mass | Kilograms (kg) | How much matter an object contains |
| \(\vec{a}\) | Acceleration | m/s² | Rate of change of velocity |
| \(\vec{v}\) | Velocity | m/s | Speed and direction of motion |
| \(p\) | Momentum | kg·m/s | Mass times velocity |
| \(t\) | Time | Seconds (s) | Duration |
Worked Examples¶
Example 1: Pushing a Shopping Cart¶
You push a 20 kg shopping cart with a force of 40 N. What acceleration does it experience?
Starting from rest, after 3 seconds: \(v = at = 2 \times 3 = 6\) m/s (about 21 km/h). That's a fast cart — which is why you don't push that hard for that long.
Example 2: Why Heavy Trucks Brake Slowly¶
A car (1,500 kg) and a truck (15,000 kg) both brake with the same force of 15,000 N.
Car:
Truck:
The car decelerates 10 times faster. This is exactly why trucks need much longer stopping distances and why tailgating a truck is dangerous.
Example 3: The Third Law in Action — A Rifle's Recoil¶
A 4 kg rifle fires a 0.01 kg bullet at 800 m/s. By the third law, the force on the bullet equals the force on the rifle (in opposite direction). By the second law, the rifle's recoil velocity is:
Momentum before: \(0\) (everything at rest) Momentum after: \(m_{bullet} \cdot v_{bullet} + m_{rifle} \cdot v_{rifle} = 0\)
The bullet goes forward at 800 m/s, the rifle kicks back at 2 m/s. Same force, same time, but the rifle is 400 times heavier — so it moves 400 times slower. That's why the recoil is a kick, not a launch.
Why \(F = ma\) and Not Something Else?¶
Why not \(F = mv\)? Or \(F = ma^2\)?
\(F = mv\) would mean that maintaining a constant velocity requires a constant force. This is Aristotle's physics — and it's wrong. A hockey puck on ice keeps moving without any force. Force causes change in velocity, not velocity itself.
\(F = ma^2\) would mean that doubling the acceleration requires only \(\sqrt{2}\) times the force. This would make the relationship between force and acceleration nonlinear and would violate basic experimental observations. When you push twice as hard, you get twice the acceleration. Every experiment confirms this linearity.
\(F = ma\) is the simplest relationship that matches reality: force is proportional to the rate of change of motion, not to motion itself.
Common Mistakes¶
- "Objects in motion slow down naturally": No. They slow down because of friction, air resistance, or other forces. In space, a thrown object moves at the same speed forever. Aristotle's intuition was wrong.
- "Heavier objects fall faster": Newton's second law with gravity gives \(F = mg\), so \(a = g\) — the mass cancels. All objects fall at the same rate (in vacuum). Galileo demonstrated this, and Newton's laws explain why.
- "The third law means forces cancel out": The action and reaction act on different objects. When you push a wall, you push the wall and the wall pushes you. These forces don't cancel because they're on different things. They would cancel if they were on the same object.
- "F = ma is always valid": It breaks down at speeds approaching light (need special relativity) and at atomic scales (need quantum mechanics). It's the low-speed, large-scale approximation — which covers 99.99% of everyday life.
Related Formulas¶
- Kinetic Energy — derived directly from \(F = ma\) and the definition of work
- Gravitational Potential Energy — what happens when the force is gravity
- Archimedes' Principle — Newton's laws applied to fluids
History¶
- ~350 BCE — Aristotle teaches that force is needed to maintain motion; rest is the natural state
- 1543 — Copernicus places the Sun at the center, implying the Earth moves (but can't explain why we don't feel it)
- 1632 — Galileo publishes Dialogue Concerning the Two Chief World Systems, introducing the concept of inertia
- 1665-66 — Newton, isolated during the plague, develops the foundations of calculus and mechanics
- 1687 — Newton publishes the Principia, containing the three laws of motion and universal gravitation
- 1905 — Einstein's special relativity shows that \(F = ma\) is an approximation valid only at speeds much less than light
References¶
- Newton, I. (1687). Philosophiæ Naturalis Principia Mathematica
- Galilei, G. (1632). Dialogue Concerning the Two Chief World Systems
- Cohen, I. B. "Newton's Laws of Motion." The Cambridge Companion to Newton, 2002.
- Feynman, R. P. The Feynman Lectures on Physics, Vol. 1, Ch. 9-12.