High School · Physics · Unit 1

Newton's Three Laws of Motion

Three short rules that explain seatbelts, shopping carts and rockets.

The bus brakes hard and your whole body tips forward, even though nobody pushed you. Why? In 1687 Isaac Newton published three laws of motion that answer that question, and almost every question like it.

On this page you'll learn each law in plain words, see it in diagrams and everyday life, work through solved problems, try an F = ma calculator, read how the laws were first published, and finish with a quiz.

Start with the first law Jump to the quiz

Key words first

Force

A push or a pull on an object. Forces have a size and a direction, so we draw them as arrows. Force is measured in newtons (N).

Net force

The total of all the forces on an object, once you account for their directions. A 50 N push to the right and 20 N of friction to the left make a net force of 30 N to the right.

Mass

How much matter is in an object, measured in kilograms (kg). Mass is also a measure of how hard it is to change the object's motion.

Acceleration

How fast velocity changes, in meters per second per second (m/s²). Speeding up, slowing down and turning are all acceleration.

Law 1: Inertia, or "things keep doing what they're doing"

"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."Newton, Principia, Law I, in Andrew Motte's English translation (1846 American edition).

In modern words: an object at rest stays at rest, and an object moving in a straight line at a steady speed keeps doing that, unless a net force acts on it.

This holds when you measure motion from a frame that isn't itself speeding up, slowing down or turning, called an inertial frame. Inside a braking bus or car, you seem to be thrown forward with no force at all, because the vehicle itself is decelerating.

This tendency to resist changes in motion is called inertia. The more mass something has, the more inertia it has. That's why a loaded truck is harder to start and stop than a bicycle.

Back to the bus. When the bus brakes, the brakes push on the bus, not on you. Your body keeps moving forward at the old speed until the seat, a pole or a seatbelt pushes back on you. Seatbelts and headrests exist because of the first law.

The brakes slow the car. Only the seatbelt (orange) can slow the passenger.

Where you see the first law

Tablecloth trick: yank a smooth cloth out fast and the dishes barely move. The short tug of friction isn't enough to get much mass moving.
Hammer head: tap the handle end down on a hard floor and the loose head slides further onto the handle. The handle stops; the head keeps going.
Spacecraft: far from planets, with almost nothing to slow it down, a probe can coast for years with its engines off.

Common mistake: "Moving things naturally slow down." A rolling ball does stop, but that's because friction and air resistance are forces pushing against it. Take those forces away and it would keep rolling. Newton himself pointed out that projectiles keep moving "so far as they are not retarded by the resistance of the air, or impelled downwards by the force of gravity."

Law 2: F = ma, or "how much push for how much change"

"The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed."Newton, Principia, Law II, Motte translation.

Today we usually write the second law as an equation:

F_net = m × a

The net force on an object equals its mass times its acceleration, and the acceleration points the same way as the net force. Rearranged, a = F_net ÷ m. That tells you two things:

More force, more acceleration. Push an empty cart twice as hard and it speeds up twice as fast.
More mass, less acceleration. Push a full cart with the same force and it speeds up more slowly.

The unit of force comes straight from this law. One newton is the force that gives a 1 kg mass an acceleration of 1 m/s². So 1 N = 1 kg·m/s².

A free-body diagram shows every force on one object as an arrow. Add them up by direction to get the net force. (This box is Example 3 below; arrows not to scale.)

Mass and weight are not the same thing

Your mass is the same everywhere. Your weight is the force of gravity on you, so it depends on where you are. Near Earth's surface, things in free fall accelerate at about 9.8 m/s² (written g), so the second law gives:

W = m × g

A 60 kg student weighs about 60 × 9.8 = 588 N on Earth. On the Moon, where g is about 1.62 m/s², the same student weighs only about 97 N, but still has a mass of 60 kg.

Where you see the second law

Shopping carts: the same shove gets an empty cart rolling much faster than a full one.
Sports: a baseball (about 0.145 kg) flies off the bat far faster than a bowling ball would from the same swing, because its mass is so small.
Car design: a lighter car can accelerate faster with the same engine force, which is one reason engineers care about weight.

Law 3: Action and reaction, or "you can't touch without being touched"

"To every action there is always opposed an equal reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts."Newton, Principia, Law III, Motte translation.

In modern words: when object A pushes or pulls on object B, B pushes or pulls back on A with a force of the same size in the opposite direction. Forces always come in these pairs.

The key detail: the two forces act on different objects. That's why they don't cancel out. When two skaters push apart, each feels a push, and each moves.

Newton gave his own examples: "If you press a stone with your finger, the finger is also pressed by the stone." And a horse pulling a stone on a rope "will be equally drawn back towards the stone."

Every force has a partner. The partners act on different objects.

Where you see the third law

Walking: your shoe pushes backward on the ground; the ground pushes you forward. On slick ice there's little friction to push against, so walking gets hard.
Swimming and rowing: you push water backward and the water pushes you forward.
Rockets: the engine forces hot gas out the back at high speed, and the gas pushes the rocket the other way. No air is needed to push against, which is why rockets work in space.
Recoil: a fire hose kicks back against the firefighter holding it.

Puzzle: if the forces are equal, why does a truck flatten a mosquito? In a crash, the truck pushes on the mosquito exactly as hard as the mosquito pushes on the truck. But by the second law, a = F ÷ m: the same force on a tiny mass gives a huge acceleration. The truck barely notices.

Worked examples

Try each one on paper first, then open the solution. We use g = 9.8 m/s² and round sensibly.

1. Pushing a cart

A 12 kg shopping cart has a net force of 30 N on it. What is its acceleration?

Show solution

a = F ÷ m = 30 N ÷ 12 kg = 2.5 m/s²

2. Speeding up a car

A 1,200 kg car speeds up at 3.0 m/s². What net force acts on it?

Show solution

F = m × a = 1,200 kg × 3.0 m/s² = 3,600 N forward

3. Two forces at once

You push a 15 kg box to the right with 50 N. Friction pushes back with 20 N. What is the box's acceleration? (See the free-body diagram above.)

Show solution

Net force = 50 N − 20 N = 30 N to the right
a = 30 N ÷ 15 kg = 2.0 m/s² to the right

The weight (15 × 9.8 = 147 N) and the floor's normal force cancel, so there's no up-or-down acceleration.

4. Weight on Earth and the Moon

A student has a mass of 60 kg. What is their weight on Earth, and on the Moon (g ≈ 1.62 m/s²)?

Show solution

Earth: W = 60 × 9.8 = 588 N
Moon: W = 60 × 1.62 ≈ 97 N

The mass is still 60 kg in both places.

5. Skaters push apart

Skater A (50 kg) and skater B (75 kg) stand still on ice and push off each other. A pushes B with 150 N. What is each skater's acceleration?

Show solution

Third law: B pushes A with 150 N the other way.
A: a = 150 ÷ 50 = 3.0 m/s²
B: a = 150 ÷ 75 = 2.0 m/s², in the opposite direction

Same force, but the lighter skater accelerates more.

6. Steady speed

A car cruises at a steady 25 m/s on a straight, level road. What is the net force on it?

Show solution

No change in speed or direction means a = 0, so F_net = m × 0 = 0 N.

The engine's forward force exactly balances air resistance and friction. That's the first law at work.

Try it: the F = ma calculator

Solve for any one of F, m or a

Things to try

Keep the force at 30 N and double the mass from 12 kg to 24 kg. What happens to the acceleration?
Find the force needed to give a 0.145 kg baseball an acceleration of 1,000 m/s².
What mass speeds up at 9.8 m/s² when the net force is 588 N? (Look back at Example 4.)

How do we know?

Newton didn't just state the laws. He tested them. In the Principia he describes hanging two balls from strings about ten feet long, letting them swing down and collide, and measuring how far each bounced. Across many trials with equal and unequal balls, he reported he "found always, without an error of 3 inches," that the changes in their motions were equal and opposite: action equaled reaction.

He also noted that the first two laws were consistent with Galileo's earlier discoveries that falling bodies cover distances in proportion to the square of the time, and that projectiles follow curved paths called parabolas.

You can repeat the spirit of these tests yourself. Physics classrooms use low-friction carts and air tracks, where a cart glides almost without slowing (law 1), speeds up more under a bigger pull (law 2), and two carts pushed apart by a spring fly off in opposite directions (law 3).

Where the laws stop working. Newton's laws are extremely accurate for everyday speeds and sizes, from baseballs to planets. For things moving close to the speed of light, physicists use Einstein's relativity, and for atoms and smaller they use quantum mechanics. Engineers still use Newton's laws for cars, bridges and most spaceflight.

A short history: the Principia (1687)

Newton's three laws first appeared in print in his book Philosophiæ Naturalis Principia Mathematica, Latin for "Mathematical Principles of Natural Philosophy." People usually just call it the Principia. Written in Latin, it uses the laws of motion together with the law of universal gravitation to explain the orbits of the planets and comets, the Moon's motion and the tides.

The book exists partly thanks to the astronomer Edmond Halley (of Halley's Comet). He visited Newton in Cambridge in 1684 to ask about planetary orbits, encouraged him to write up his work, edited it, and saw it through the printer. The Royal Society agreed to publish it, but it had already spent its money on an illustrated book about fish, so Halley paid for the printing himself.

The title page carries an approval by the Society's president, the diarist Samuel Pepys. On 5 July 1687, Halley wrote to tell Newton the printing was finished.

An original drawing for this page, not a scan of the real book.

1684 Halley visits Newton in Cambridge to ask about planetary orbits; by November Newton sends him a short paper on motion.
1686 The Royal Society receives Book I and decides to publish. Halley takes charge of printing at his own expense.
1687 The Principia is published in Latin. On 5 July Halley tells Newton it's finished.
1713 and 1726 Newton publishes revised second and third editions.
1729 Andrew Motte's English translation appears, two years after Newton's death in 1727.
1750–1752 Leonhard Euler writes the second law in the general form F = ma that students use today. Newton stated it in words, about "change of motion."

Test yourself

The Force Check

Talk about it

You're standing on a skateboard and throw a heavy backpack forward. What happens to you, and which law explains it?
A horse pulls a cart, and the cart pulls back on the horse just as hard. So how does anything ever move? (Hint: think about which forces act on the horse alone.)
Astronauts on the International Space Station feel "weightless." Do they still have mass? Do they still have inertia? How could you tell?
Newton wrote the second law in words about "change of motion," and Euler later wrote F = ma. Why might a simple equation make an idea easier to use?

Sources

Isaac Newton, The Mathematical Principles of Natural Philosophy, trans. Andrew Motte (1729), first American edition (New York: Daniel Adee, 1846), Axioms, or Laws of Motion, and Scholium. Project Gutenberg eBook #76404. https://www.gutenberg.org/ebooks/76404
OpenStax, Physics (high school), Chapter 4 "Forces and Newton's Laws of Motion": 4.1 Force, 4.2 Newton's First Law of Motion: Inertia, 4.3 Newton's Second Law of Motion, 4.4 Newton's Third Law of Motion.
NIST, "Metric (SI) units" (the newton, 1 N = 1 kg·m/s²). https://www.nist.gov/pml/owm/metric-si/si-units
NIST CODATA, "Standard acceleration of gravity" (9.80665 m/s², rounded to 9.8 on this page). https://physics.nist.gov/cgi-bin/cuu/Value?gn
NASA Goddard Space Flight Center, "Moon Fact Sheet" (surface gravity 1.62 m/s²). https://nssdc.gsfc.nasa.gov/planetary/factsheet/moonfact.html
Desmond King-Hele, "Introductory remarks," Notes and Records of the Royal Society 42, no. 1 (1988): 3, doi:10.1098/rsnr.1988.0002, on Pepys's imprimatur, Halley's financing and his letter of 5 July 1687. https://royalsocietypublishing.org/rsnr/article/42/1/3/55795/Introductory-remarks
Cambridge University Library, "Footprints of the Lion" exhibition catalogue, Halley's 1684 visit, De motu and the 1686 decision to publish. https://wwwe.lib.cam.ac.uk/CUL/exhibitions/Footprints_of_the_Lion/glory.pdf
Pembroke College, Oxford, "#26 Philosophiae Naturalis Principia Mathematica" (the Royal Society's spent budget and De Historia Piscium). https://www.pmb.ox.ac.uk/26-philosophiae-naturalis-principia-mathematica
The Euler Archive (University of the Pacific), E177, "Découverte d'un nouveau principe de mécanique" (written 1750, published 1752; first general appearance of F = ma). https://scholarlycommons.pacific.edu/euler-works/177/

Credits: all diagrams on this page, including the book drawing and the share image, are original SVG made for Jugami. Quotations are from the public-domain Motte translation of the Principia.