Laws of Motion: _{Three statements that make powerful predictions}

Imagine a soccer match. You’re sitting down, ready to watch your favorite team face its rival. The referee flips a coin; your team wins the coin toss and chooses to take the kickoff. The referee places the ball in the center circle, and players line up on either side. The ball sits there, not moving, as the fans in the stadium start to cheer and the players take their positions (see figure 1). Finally, the referee blows the whistle, and your team’s star forward kicks the ball to a teammate downfield. The game begins! You watch as the ball changes direction with each pass from one player’s foot to another. Then—bam!—two players collide when one attempts a block. The players end up on the ground while the ball rolls off the field.

Physics and the laws of motion are probably not on your mind while you watch a soccer match, and you do not need to solve mathematical equations to determine how your team scored a goal. Yet the ball’s motion (and, to some extent, the players’ motion) in a soccer game can be described with fundamental physical laws and quantified through measurement and calculation. Players can use these laws to make decisions that will influence the game's outcome.

Isaac Newton was not playing soccer when he started thinking about the physics of motion as a student at Trinity College at Cambridge University in the 1660s. In fact, he often forgot to get any exercise at all. Instead, he spent many hours by himself, reading about Galileo’s experiments, Descartes’ philosophy of action, and Kepler’s observations of planetary orbits. Unlike college students today, Newton had no classes and was expected to learn entirely on his own.

Which he did. Newton read, and he wrote. He wrote what was then known about motion and the conclusions that Galileo, Descartes, Wren, Kepler, and others had reached. He also wrote his own ideas, questions, and suggestions for how to answer his questions. His notebooks from the 1660s are filled with questions, ideas, and equations. Those ideas included prototypes—or initial ideas—for his laws of motion, but he would not publish those ideas for more than 20 years.

Newton finally did publish his laws of motion—which he called axioms—in Principia Mathematica in 1687. He used examples of projectiles, comets, and horse-drawn plows, explaining how these laws applied to all of them. Newton’s insights built on those of other scientists and provided a foundation for modern physics.

In this module, we’ll learn more about how Newton developed his laws of motion and how they can help us understand everything from a soccer match to the solar system.

In his first axiom, Newton wrote, “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.” Translated into more modern language, Newton’s First Law might read: If an object is at rest, it remains at rest unless it is acted on by an external force. If an object is in motion, it remains in motion at a constant velocity (including both speed and direction of motion) unless it is acted on by an external force.

How does this law apply to our soccer game? When the referee places the ball in the middle of the field, you can say that the ball is at rest, as shown in Figure 1. It will stay there on the field, not moving, until someone kicks it or the wind blows and pushes it. When a player kicks the ball, they apply a force that moves the ball in the direction that they kicked it, as shown in Figure 2.

The ball does not keep going in a straight line, though, as Newton suggests. If it did, it would fly out of the stadium and into space. This does not mean Newton was wrong. Instead, it means that other forces are acting on the ball. So, what are these other forces acting on it to “compel it to change” that state of motion?

• Gravity is a downward force acting on the ball, pushing it back towards the ground.
• Air resistance is a frictional force between the ball and the air that slows the ball’s motion.
• The player may also have put “spin” on the ball, causing it to rotate on an axis and creating differential air pressure that curves the ball’s motion.

All three forces work together to cause the ball to follow an arcuate path back to the ground down the field.

Other forces can act to change the ball’s motion, too. For example, another player could jump to block the ball, or the wind could push the ball while it is in the air. Newton used the example of a projectile being slowed by air resistance and brought to the ground by gravity. He also extended his example to the planets in orbit around the sun, which would continue in their paths for much longer because they faced “less resistance in more free spaces.” He used these examples to highlight that the forces that act on an object to change its state of motion can be visible (a soccer player’s foot, a cannon to shoot a projectile) or invisible (gravity, air resistance).

Today, we say that Newton’s first law defines the concept of “inertia,” even though he did not use that word. Inertia is the tendency of an object to stay in its state of motion—at rest or moving—until it is acted on by another force. Over time, the concept of inertia has moved from physics to popular culture, and you might use it to describe your tendency to remain on the couch watching a soccer game. But Newton’s simple axiom was revolutionary because it brought together observations about the motion of planets, projectiles, and balls rolling down inclined planes, and today, it can be applied to every aspect of our lives.

In his second axiom, Newton wrote, “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.” He followed this statement by saying, “If any force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively.”

Newton was describing what we would today call direct proportionality, which we can write as a mathematical equation. Newton mentions “direction,” noting that motion will happen in the same direction as the impressed or applied force. Here, he is describing what we now call vector quantities, which include both magnitude and direction. Both force and acceleration are vector quantities. In contrast, mass is a scalar quantity, meaning it has a magnitude but no direction.

Think again of your soccer team after the kickoff. A player gains control of the ball and looks for a teammate to pass to. They can tap the ball gently to a teammate on their left or kick it to a teammate on their right, further down the field. In each case, the force will differ in both magnitude (gentle tap vs. aggressive kick) and direction (to the left vs. the right), but the ball's mass stays the same.

Today, this proportional relationship is most commonly expressed as a mathematical equation (Equation 1):

This equation tells us that a larger force acting on the same mass will produce a proportionally larger acceleration, as Newton wrote.

In our soccer match, the ball’s mass is a known scalar quantity and is a constant. As a player approaches the ball, they may be looking for their teammates and thinking, "Which direction should I send it?” and "How hard should I kick it?” In other words, they are intuitively considering the force vector they will apply. They could run fast toward the ball, then kick it lightly to a teammate on their left. Or they could kick it and run with it, continuing to dribble down the field and accelerating the ball. Eventually, they may apply a larger force to send it over their opponents’ heads and into the goal.

Direct proportionality is a powerful tool for prediction. For example, it tells us that if the soccer ball’s mass changed—if the ball were wet, for example—the same applied force would produce a smaller acceleration.

In his third axiom, Newton wrote, “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.” He continued with an example: “If you press a stone with your finger, the finger is also pressed by the stone.”

In a later section of the Principia, Newton describes a series of experiments he conducted with pendulums, building on the work of others, including Christian Huygens, a Dutch scientist who had worked on building an accurate pendulum. Newton built on Huygens’ work, experimenting with two spherical bodies hanging as pendulums, as illustrated in Figure 3.

Newton described the general concept of his experiments using the annotations in Figure 3, such as, “Bring the body A to any point R on the arc EAF, and … let it go from thence…” He described in detail how to quantify the motion of a single pendulum and how each pendulum would influence the other when they collided.

He continued, “Thus trying the thing with pendulums of ten feet, in unequal as well as equal bodies, and making the bodies to concur after a descent through large spaces, as of 8, 12, or 16 feet, I found always, without an error of 3 inches, that when the bodies concurred together directly, equal changes towards the contrary parts were produced in their motions, and of consequence, that the action and reaction were always equal.”

In other words, Newton tested pendulums of different lengths—up to 16 feet long—and different sizes. The experiments required very large, tall rooms so the pendulums could swing freely. He described the combination of the pendulum’s mass and distance traveled as “parts of motion,” which we might think of today as “momentum.” In every experiment and with every combination, the action and reaction were the same: When one pendulum hit another, the second rose to the same level as the first pendulum had been released from.

Although pendulums are what Newton experimented with, he saw that the law applied far beyond this experiment. In fact, it applied when there was no motion at all, as he illustrated in his primary example, “If you press a stone with your finger, the stone presses back.”

If we go back to our soccer game, the idea of “the stone pressing back” corresponds to the game’s start, when the ball is sitting on the field (Figure 1). The ball exerts a force on the ground, and the ground exerts a force on the ball. Those forces must be equal in magnitude and opposite in direction, or else the ball or the ground would move.

Newton’s third law is often written today as “for every action, there is an equal and opposite reaction.” That can make people think there must be motion for this law to apply. For example, you might be inclined to think about the player who jumps up to block a pass with their head (Figure 4). The ball strikes their head and flattens a bit before changing direction. The ball's action transfers to the player's head, causing (unfortunately) the brain to move backward in the skull, sometimes resulting in a concussion, kind of like Newton’s pendulum.

Heading a soccer ball is actually a complex example that illustrates all three of Newton’s laws:

1. The ball’s inertia as it strikes a player’s head transfers motion to the skull.
2. The ball’s force is equal to its mass times its acceleration.
3. As the player pushed off the Earth to jump into the air, the Earth moved a tiny amount beneath her.

One of Newton’s great insights is that motion could be fully described with only three laws. His work provided the foundation of what is now called classical mechanics. Classical mechanics and the laws of motion do a great job of explaining motion in the world around us: the arc of a kicked soccer ball, the rebound when the ball hits the goalpost, and why kicking the ball harder makes it go farther.

However, Newton’s laws fail to explain the behavior of objects at very small scales—atomic and subatomic scales. The field of physics that deals with these scales is quantum mechanics, which developed in the early 1900s as a number of physicists made observations that classical mechanics could not explain.

At around the same time, Albert Einstein recognized another limitation of the laws of motion. As part of his theory of special relativity, which he published in 1905, Einstein postulated that there is no such thing as “at rest.” In other words, everything is in motion, and you are only perceiving the motion relative to you. The theory of special relativity doesn’t affect the physics of our soccer game, but it does mean that the entire game is moving through space.

It’s probably a safe bet that your favorite soccer player is not thinking about the equation F = ma when they prepare to pass the ball to a teammate or take a shot on goal. Yet it is Newton’s laws of motion that allow the teammate to determine where they should be to take the pass, and the goalie to try to block the shot. Newton articulated many concepts we now consider intuitive: your favorite soccer player is almost certainly thinking about the force they want to impart on the ball, at what angle, and what that will mean as it travels through the air. You might be making those same calculations when you casually toss a key to your friend or hit a ball with an arm or a racquet. Newton’s laws allow us to quantify the complexity of motion around us, from soccer balls to the moon, from toy cars to rockets.