How Newton's Laws of Motion Work

By: William Harris  | 
Isaac Newton portrait
Isaac Newton established classical mechanics, with his three laws on motion. He also formulated the theory of gravity, among his many other contributions to science and mathematics. Bettmann/Getty Images

Next to E = mc², F = ma is the most famous equation in all of physics. Yet many people remain mystified by this fairly simple algebraic expression. It's actually a mathematical representation of Isaac Newton's second law of motion, one of the great scientist's most important contributions. The "second" implies that other laws exist, and, luckily for students and trivia hounds everywhere, there are only two additional laws of motion. Here they are:

  1. Every object persists in its state of rest or uniform motion — in a straight line unless it is compelled to change that state by forces impressed on it.
  2. Force is equal to the change in momentum per change in time. For a constant mass, force equals mass times acceleration.
  3. For every action, there is an equal and opposite reaction.

These three laws form the foundation of what is known as classical mechanics, or the science concerned with the motion of bodies related to the forces acting on it. The bodies in motion could be large objects, such as orbiting moons or planets, or they could be ordinary objects on Earth's surface, such as moving vehicles or speeding bullets. Even bodies at rest are fair game.


Where classical mechanics begins to fall apart is when it tries to describe the motion of very small bodies, such as electrons. Physicists had to create a new paradigm, known as quantum mechanics, to describe the behavior of objects at the atomic and subatomic level.

But quantum mechanics is beyond the scope of this article. Our focus will be classical mechanics and Newton's three laws. We'll examine each in detail, both from a theoretical and a practical point of view. We'll also discuss the history of Newton's laws, because how he arrived at his conclusions is just as important as the conclusions themselves. The best place to start, of course, is at the beginning with Newton's first law.


Newton's First Law (Law of Inertia)

Marbles rolling down ramps
According to Newton's first law, the marble on that bottom ramp should just keep going. And going.

Let's restate Newton's first law in everyday terms:

An object at rest will stay at rest, forever, as long as nothing pushes or pulls on it. An object in motion will stay in motion, traveling in a straight line, forever, until a net external force pushes or pulls on it.

The "forever" part is difficult to swallow sometimes. But imagine that you have three ramps set up as shown below. Also imagine that the ramps are infinitely long and infinitely smooth. You let a marble roll down the first ramp, which is set at a slight incline. The marble speeds up on its way down the ramp.


Now, you give a gentle push to the marble going uphill on the second ramp. It slows down as it goes up. Finally, you push a marble on a ramp that represents the middle state between the first two — in other words, a ramp that is perfectly horizontal. In this case, the marble will neither slow down nor speed up. In fact, it should keep rolling. Forever.

Physicists use the term inertia to describe this tendency of an object to resist a change in its motion. The Latin root for inertia is the same root for "inert," which means lacking the ability to move. So, you can see how scientists came up with the word. What's more amazing is that they came up with the concept. Inertia isn't an immediately apparent physical property, such as length or volume. It is, however, related to an object's mass. To understand how, consider the sumo wrestler and the boy shown below.

sumo wrestler and little boy in ring
Which person in this ring will be harder to move? The sumo wrestler or the little boy?
AFP/Getty Images

Let's say the wrestler on the left has a mass of 136 kilograms, and the boy on the right has a mass of 30 kilograms (scientists measure mass in kilograms). Remember the object of sumo wrestling is to move your opponent from his position. Which person in our example would be easier to move? Common sense tells you that the boy would be easier to move, or less resistant to inertia.

You experience inertia in a moving car all the time. In fact, seat belts exist in cars specifically to counteract the effects of inertia. Imagine for a moment that a car at a test track is traveling at a speed of 55 mph (80 kph). Now imagine that a crash test dummy is inside that car, riding in the front seat. If the car slams into a wall, the dummy flies forward into the dashboard.

Why? Because, according to Newton's first law, an object in motion will remain in motion until an outside force acts on it. When the car hits the wall, the dummy keeps moving in a straight line and at a constant speed until the dashboard applies a force. Seatbelts hold dummies (and passengers) down, protecting them from their own inertia.

Interestingly, Newton wasn't the first scientist to come up with the law of inertia. That honor goes to Galileo and to René Descartes. In fact, the marble-and-ramp thought experiment described previously is credited to Galileo. Newton owed much to events and people who preceded him. Before we continue with his other two laws, let's review some of the important history that informed them.


A Brief History of Newton's Laws

Copernican world system
This illustration shows the Copernican world system. Nicolaus Copernicus was the first to propose that Earth revolved around the sun rather than the other way around. Stefano Bianchetti/CORBIS/Corbis via Getty Images

The Greek philosopher Aristotle dominated scientific thinking for many years. His views on motion were widely accepted because they seemed to support what people observed in nature. For example, Aristotle thought that weight affected falling objects. A heavier object, he argued, would reach the ground faster than a lighter object dropped at the same time from the same height. He also rejected the notion of inertia, asserting instead that a force must be constantly applied to keep something moving. Both of these concepts were wrong, but it would take many years — and several daring thinkers — to overturn them.

The first big blow to Aristotle's ideas came in the 16th century when Nicolaus Copernicus published his sun-centered model of the universe. Aristotle theorized that the sun, the moon and the planets all revolved around Earth on a set of celestial spheres. Copernicus proposed that the planets of the solar system revolved around the sun, not Earth. Although not a topic of mechanics per se, the heliocentric cosmology described by Copernicus revealed the vulnerability of Aristotle's science.


Galileo Galilei was the next to challenge the Greek philosopher's ideas. Galileo conducted two now-classic experiments that set the tone and tenor for all scientific work that would follow. In the first experiment, he dropped a cannonball and a musket ball from the Leaning Tower of Pisa. Aristotelian theory predicted that the cannonball, much more massive, would fall faster and hit the ground first. But Galileo found that the two objects fell at the same rate and struck the ground roughly at the same time.

Some historians question whether Galileo ever carried out the Pisa experiment, but he followed it with a second phase of work that has been well-documented. These experiments involved bronze balls of various sizes rolling down an inclined wood plane. Galileo recorded how far a ball would roll in each one-second interval. He found that the size of the ball didn't matter — the rate of its descent along the ramp remained constant. From this, he concluded that freely falling objects experience uniform acceleration regardless of mass, as long as extraneous forces, such as air resistance and friction, can be minimized.

But it was René Descartes, the great French philosopher, who would add new depth and dimension to inertial motion. In his "Principles of Philosophy," Descartes proposed three laws of nature. The first law states that each thing, as far as is in its power, always remains in the same state; and that consequently, when it is once moved, it always continues to move. The second holds that all movement is, of itself, along straight lines. This is Newton's first law, clearly stated in a book published in 1644 — when Newton was still a newborn!

Clearly, Isaac Newton studied Descartes. He put that studying to good use as he single-handedly launched the modern era of scientific thinking. Newton's work in mathematics resulted in integral and differential calculus. His work in optics led to the first reflecting telescope. And yet his most famous contribution came in the form of three relatively simple laws that could be used, with great predictive power, to describe the motion of objects on Earth and in the heavens. The first of these laws came directly from Descartes, but the remaining two belong to Newton alone.

He described all three in "The Mathematical Principles of Natural Philosophy," or the Principia, which was published in 1687. Today, the Principia remains one of the most influential books in the history of human existence. Much of its importance lies within the elegantly simple second law, F = ma, which is the topic of the next section.


Newton's Second Law (Law of Motion)

One dog pulling a sled, illustrating f = ma
If you want to calculate the acceleration, first you need to modify the force equation to get a = F/m. When you plug in the numbers for force (100 N) and mass (50 kilograms), you find that the acceleration is 2 m/s2.

You may be surprised to learn that Newton wasn't the genius behind the law of inertia. But Newton himself wrote that he was able to see so far only because he stood on "the shoulders of Giants." And see far he did. Although the law of inertia identified forces as the actions required to stop or start motion, it didn't quantify those forces. Newton's second law supplied the missing link by relating force to acceleration. This is what it said:

When a force acts on an object, the object accelerates in the direction of the force. If the mass of an object is held constant, increasing force will increase acceleration. If the force on an object remains constant, increasing mass will decrease acceleration. In other words, force and acceleration are directly proportional, while mass and acceleration are inversely proportional.

Technically, Newton equated force to the differential change in momentum per unit time. Momentum is a characteristic of a moving body determined by the product of the body's mass and velocity. To determine the differential change in momentum per unit time, Newton developed a new type of math — differential calculus. His original equation looked something like this:


F = (m)(Δv/Δt)

where the delta symbols signify change. Because acceleration is defined as the instantaneous change in velocity in an instant of time (Δv/Δt), the equation is often rewritten as:

F = ma

The F, the m and the a in Newton's formula are very important concepts in mechanics. The F is force, a push or pull exerted on an object. The m is mass, a measure of how much matter is in an object. And the a is acceleration, which describes how an object's velocity changes over time. Velocity, which is similar to speed, is the distance an object travels in a certain amount of time.

The equation form of Newton's second law allows us to specify a unit of measurement for force. Because the standard unit of mass is the kilogram (kg) and the standard unit of acceleration is meters per second squared (m/s2), the unit for force must be a product of the two — (kg)(m/s2). This is a little awkward, so scientists decided to use a Newton as the official unit of force. One Newton, or N, is equivalent to 1 kilogram-meter per second squared. There are 4.448 N in 1 pound.

Dog pulling a sled, illustrating the f = ma equation
Notice that doubling the force by adding another dog doubles the acceleration.

So, what can you do with Newton's second law? As it turns out, F = ma lets you quantify motion of every variety. Let's say, for example, you want to calculate the acceleration of the dog sled shown at left.

Now let's say that the mass of the sled stays at 50 kilograms and that another dog is added to the team. If we assume the second dog pulls with equal force to the first (100 N), the total force would be 200 N and the acceleration would be 4 m/s2. However, doubling the mass to 100 kilograms would halve the acceleration to 2 m/s2.

Four dogs pulling a sled, illustrating the f = ma equation
If two dogs are on each side, then the total force pulling to the left (200 N) balances the total force pulling to the right (200 N). Since the net force on the sled is zero, the sled doesn't move.

Finally, let's imagine that a second dog team is attached to the sled so that it can pull in the opposite direction.

This is important because Newton's second law is concerned with net forces. We could rewrite the law to say: When a net force acts on an object, the object accelerates in the direction of the net force.

Now imagine that one of the dogs on the left breaks free and runs away. Suddenly, the force pulling to the right is larger than the force pulling to the left, so the sled accelerates to the right.

What's not so obvious in our examples is that the sled is also applying a force on the dogs. In other words, all forces act in pairs. This is Newton's third law — and the topic of the next section.


Newton's Third Law (Law of Force Pairs)

Katinka Hosszu , backstroke race push off
Katinka Hosszu of Hungary pushes off at the start of the women's 50 meter backstroke race during the FINA Swimming World Cup in Dubai, 2013. That's one heck of a force! MARWAN NAAMANI/AFP via Getty Images

Newton's third law is probably the most familiar. Everyone knows that every action has an equal and opposite reaction, right? Unfortunately, this statement lacks some necessary detail. This is a better way to say it:

A force is exerted by one object on another object. In other words, every force involves the interaction of two objects. When one object exerts a force on a second object, the second object also exerts a force on the first object. The two forces are equal in strength and oriented in opposite directions.

Many people have trouble visualizing this law because it's not as intuitive. In fact, the best way to discuss the law of force pairs is by presenting examples. Let's start by considering a swimmer facing the wall of a pool. If she places her feet on the wall and pushes hard, what happens? She shoots backward, away from the wall.


Clearly, the swimmer is applying a force to the wall, but her motion indicates that a force is being applied to her, too. This force comes from the wall, and it's equal in magnitude and opposite in direction.

Next, think about a book lying on a table. What forces are acting on it? One big force is Earth's gravity. In fact, the book's weight is a measurement of Earth's gravitational attraction. So, if we say the book weighs 10 N, what we're really saying is that Earth is applying a force of 10 N on the book. The force is directed straight down, toward the center of the planet. Despite this force, the book remains motionless, which can only mean one thing: There must be another force, equal to 10 N, pushing upward. That equal and opposite force is coming from the table.

If you're catching on to Newton's third law, you should have noticed another force pair described in the paragraph above. Earth is applying a force on the book, so the book must be applying a force on Earth. Is that possible? Yes, it is, but the book is so small that it cannot appreciably accelerate something as large as a planet.

You see something similar, although on a much smaller scale, when a baseball bat strikes a ball. There's no doubt the bat applies a force to the ball: It accelerates rapidly after being struck. But the ball must also be applying a force to the bat. The mass of the ball, however, is small compared to the mass of the bat, which includes the batter attached to the end of it. Still, if you've ever seen a wooden baseball bat break into pieces as it strikes a ball, then you've seen firsthand evidence of the ball's force.

These examples don't show a practical application of Newton's third law. Is there a way to put force pairs to good use? Jet propulsion is one application. Used by animals such as squid and octopuses, as well as by certain airplanes and rockets, jet propulsion involves forcing a substance through an opening at high speed. In squid and octopuses, the substance is seawater, which is sucked in through the mantle and ejected through a siphon. Because the animal exerts a force on the water jet, the water jet exerts a force on the animal, causing it to move. A similar principle is at work in turbine-equipped jet planes and rockets in space.

Speaking of outer space, Newton's other laws apply there, too. By using his laws to analyze the motion of planets in space, Newton was able to come up with a universal law of gravitation.


Applications and Limitations of Newton's Laws

Digital generated image of earth rising
Earth rises above the moonscape. Newton reasoned that the moon moves around Earth in the same way that a stone whirls around the end of a string. xia yuan/Getty Images

By themselves, the three laws of motion are a crowning achievement, but Newton didn't stop there. He took those ideas and applied them to a problem that had stumped scientists for years: the motion of planets. Copernicus placed the sun at the center of a family of orbiting planets and moons, while the German astronomer Johannes Kepler proved that the shape of planetary orbits was elliptical, not circular. But no one had been able to explain the mechanics behind this motion. Then, as the story goes, Newton saw an apple fall to the ground and was seized by inspiration. Could a falling apple be related to a revolving planet or moon? Newton believed so. This was his thought process to prove it:

  1. An apple falling to the ground must be under the influence of a force, according to his second law. That force is gravity, which causes the apple to accelerate toward Earth's center.
  2. Newton reasoned that the moon might be under the influence of Earth's gravity, as well, but he had to explain why the moon didn't fall into Earth. Unlike the falling apple, it moved parallel to Earth's surface.
  3. What if, he wondered, the moon moved about Earth in the same way as a stone whirled around at the end of a string? If the holder of the string let go — and therefore stopped applying a force — the stone would obey the law of inertia and continue traveling in a straight line, like a tangent extending from the circumference of the circle.
  4. But if the holder of the string didn't let go, the stone would travel in a circular path, like the face of a clock. In one instant, the stone would be at 12 o'clock. In the next, it would be at 3 o'clock. A force is required to pull the stone inward so it continues its circular path or orbit. The force comes from the holder of the string.
  5. Next, Newton reasoned that the moon orbiting Earth was the same as the stone whirling around on its string. Earth behaved as the holder of the string, exerting an inward-directed force on the moon. This force was balanced by the moon's inertia, which tried to keep the moon moving in a straight-line tangent to the circular path.
  6. Finally, Newton extended this line of reasoning to any of the planets revolving around the sun. Each planet has inertial motion balanced by a gravitational attraction coming from the center of the sun.

It was a stunning insight — one that eventually led to the universal law of gravitation. According to this law, any two objects in the universe attract each other with a force that depends on two things: the masses of the interacting objects and the distance between them. More massive objects have bigger gravitational attractions. Distance diminishes this attraction. Newton expressed this mathematically in this equation:


F = G(m1m2/r2)

where F is the force of gravity between masses m1 and m2, G is a universal constant and r is the distance between the centers of both masses.

Over the years, scientists in just about every discipline have tested Newton's laws of motion and found them to be amazingly predictive and reliable. But there are two instances where Newtonian physics break down. The first involves objects traveling at or near the speed of light. The second problem comes when Newton's laws are applied to very small objects, such as atoms or subatomic particles that fall in the realm of quantum mechanics.


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