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Momentum Moving objects have a quantity that objects at rest don’t have. More than a hundred years ago, this quantity was called impedo. A boulder at rest had no impedo, while the same boulder rolling down a steep incline possessed impedo. The faster it moved, the greater the impedo. The change in impedo depended on force and, more importantly, on how long the force acted. Apply a force to a cart and you give it impedo. Apply a long force and you give it more impedo. But what do we mean by “long?” Does “long” refer to time or distance?

When this distinction was made, the term impedo gave way to two more precise ideas momentum and kinetic energy. And these two ideas are related to a cluster of other concepts including work, power, efficiency, potential energy, and impulse. What are the distinctions and relationships among these quantities?

How can they be used to analyze matter in motion, the workings of machines, and even such complex phenomena as the energy sources that power modern industry and the “machinery” of living organisms?

We all know that a heavy truck is more difficult to stop than a small car moving at the same speed. We state this fact by saying that the truck has more momentum than the car. By momentum, we mean inertia in motion or, more specifically, the product of the mass of an object and its velocity; that is,

Momentum = mv

Or, in shorthand notation,

Momentum = mass x velocity

When direction is not an important factor, we can say momentum = mass x speed, which we still abbreviate mv. We can see from the definition that a moving object can have a large momentum if either its mass or its velocity is large or both its mass and its velocity are large. A truck has more momentum than a car moving at the same velocity because it has a greater mass. We can see that a huge ship moving at a small velocity can have a large momentum, as can a small bullet moving at a high velocity. A massive truck moving down a steep hill with no brakes has a large momentum, whereas the same truck at rest has no momentum at all.

Changes in momentum may occur when there is a change in the mass of an object, or a change in its velocity, or both. If momentum changes while the mass remains unchanged, as is most often the case, then the velocity changes. Acceleration occurs. And what produces acceleration? The answer is force. The greater the force acting on an object, the greater will be the change in velocity and, hence, the change in momentum.

But something else is important also: time how long the force acts. Apply a force briefly to a stalled automobile and you produce a small change in its momentum. Apply the same force over an extended period of time, and a greater momentum change results (Figure4.2). A long sustained force produces

more change in momentum than the same force applied briefly. So, for changing an object’s momentum, both force and the time interval during which the

force acts are important. impulse. The quantity “force x time interval” is called impuls

If you wish to produce the maximum increase in the momentum of something, you not only apply the greatest force, you also extend the time of application as

much as possible (hence the different results obtained by pushing briefly on a stalled automobile and by giving it a sustained push).

Long-range cannons have long barrels. The longer the barrel, the greater the velocity of the emerging cannonball or shell. Why? The force of exploding gunpowder in a long barrel acts on the cannonball for a longer time, increasing the impulse on it, which increases its momentum. Of course, the force that acts on the cannonball is not steady it is strong at first and weaker as the gases expand.

Most often the forces involved in impulses vary over time. The force that acts on the golf ball in Figure 4.3, for example, increases rapidly as the ball is distorted and then decreases as the ball comes up to speed and returns to its original shape. When we speak of any force that makes up impulse in this chapter, we mean the average force.

Imagine that you are in a car that is out of control, and you’re faced with a choice of slamming either into a concrete wall or into a haystack. You don’t need much physics knowledge to make the better decision, but knowing some physics aids you in knowing why hitting something soft is entirely different from hitting something hard. Whether you hit the wall or the haystack, your momentum will be decreased by the same amount, and this means that the impulse required to stop you is the same. The same impulse means the same product of force and time, not the same force or the same time. You have a choice. By hitting the haystack instead of the wall, you extend the time of impact you extend the time during which your momentum is brought to zero. The longer time is compensated for by a lesser force. If you extend the time of impact 100 times, you reduce the force of impact to a hundredth of  what it might have been. So, whenever you wish the force of an impact to be small, extend the time of the impact. And, conversely, if the time over which the force acts is short, the force itself will be comparatively large, for a given change in momentum. Going back to the example of the car, you can see why you are in much more trouble if you hit the concrete wall rather than the haystack. Your time of impact is short, so the force on you is large, as your momentum decreases to zero.

The Moving Earth | Nicolaus Copernicus - In 1543, the Polish astronomer Nicolaus Copernicus caused a great controversy when he published a book proposing that the Earth revolved around the Sun.* This idea conflicted with the popular view that the Earth was the center of the universe. Copernicus’s concept of a Sun centered solar system was the result of years of studying the planets. He had kept his theory from the public for two reasons. The first reason was that he feared persecution: 

a theory so completely different from common opinion would surely be taken as an attack on the established order. The second reason was that he had reservations about it himself: he could not reconcile the idea of a moving Earth with the prevailing ideas of motion. The concept of inertia was unknown to him and to others of his time. In the final days of his life, at the urging of close friends, he sent his manuscript, De Revolutionibus Orbium Coelestium, to the printer. The final copy of his famous exposition reached him on the day he died May 24, 1543.

The idea of a moving Earth was much debated. Europeans thought about the universe much as Aristotle had, and the existence of a force big enough to keep the Earth moving was beyond their imagination. They had no concept of inertia. One of the arguments against a moving Earth was the following:

Consider a bird sitting at rest on a branch of a tall tree. On the ground below is a fat, juicy worm. The bird sees the worm and drops vertically below and catches it. It was argued that this would be impossible if the Earth were moving. A moving Earth would have to travel at an enormous speed to circle the Sun in one year. While the bird would be in the air descending from its branch to the ground below, the worm would bswept far away along with the moving Earth. It seemed that catchin a worm on a moving Earth would be an impossible task. The fact that birds do catch worms from tree branches seemed to be clear evi dence that the Earth must be at rest. Can you see the mistake in this argument? You can if you use the concept of inertia. You see, not only is the Earth moving at a great speed, but so are the tree, the branch of the tree, the bird that sits on it, the worm below, and even the air in between. Things in motion remain in motion if no unbalanced forces are acting on them So when the bird drops from the branch, its initial sideways motion remains unchanged. It catches the worm quite unaffected by the motion of its total environment. We live on a moving Earth. If you stand next to a wall and jump up so that your feet are no longer in contact with the floor, does the moving wall slam into you? Why not? It doesn’t because you are also traveling at the same speed, before, during, and after your jump. The speed of the Earth relative to the Sun is not the speed of the wall relative to you. Four hundred years ago, people had difficulty with ideas like these. One reason is that they didn’t yet travel in high-speed vehicles. Rather, they experienced slow, bumpy rides in horse-drawn carts. People were less aware of the effects of inertia. Today, we can flip a coin in a high-speed car, bus, or plane and catch the vertically moving coin as easily as we could if the vehicle were at rest. We see evidence of the law of inertia when the horizontal motion of the coin before, during, and after the catch is the same. The coin always keeps up with us.

Newton’s Laws of Motion - A heavy parachutist falls faster than a lighter one and, there fore, has a rougher landing but why? Have you tried the party trick where you pull a tablecloth out from under place settings and the dishes stay put? How does this “trick” work, and what law of motion does it demonstrate? Have you heard the expression “You can’t touch without being touched”? Does this statement about the objective world of physics have a corollary in the world of human emotions? How did Newton’s laws get us to the Moon? How do birds fly, rockets take off, and people walk? How do Newton’s laws of motion interface with modern discoveries about motion gained from relativity and quantum mechanics?

Galileo’s work set the stage for Isaac Newton, who was born shortly after Galileo’s death in 1642. By the time Newton was 23, he had developed his famous three laws of motion, which completed the overthrow of Aristotelian ideas about motion. These three laws first appeared in one of the most famous books of all time, Newton’s PhilosophiaeNaturalisPrincipiaMathematica,*often simply known as the Principia. 

The first law is a restatement of Galileo’s concept of inertia; 

the second law relates acceleration to its cause—force; and the third is the

law of action and reaction. Newton’s first law is:

Every object continues in its state of rest, or a uniform speed in a straight line, unless acted on by a nonzero force. The key word in this law is continues; an object continues to do whatever it happens to be doing unless a force is exerted upon it. If the object is at rest, it continues in a state of rest. This is nicely demonstrated when a tablecloth is skill fully whipped from beneath dishes sitting on a tabletop, leaving the dishes in their initial state of rest (Figure 1).

On the other hand, if an object is moving, it continues to move without changing its speed or direction, as evidenced by space probes that continually move in outer space. This property of objects to resist changes in motion is called inertia (Figures1 and 2). 

CHECK YOURSELF

When a space shuttle travels in a nearly circular orbit around the Earth, is a force required to maintain its high speed? If the force of gravity were suddenly cut off, what type of path would the shuttle follow?

CHECK YOUR ANSWER

There is no force in the direction of the shuttle’s motion, which is why it coasts at a constant speed by its own inertia. The only force acting on it is the force of gravity, which acts at right angles to its motion (toward the Earth’s center). We’ll see later that this right-angled force holds the shuttle in a circular path. If it were cut off, the shuttle

would fly off in a straight line at a constant velocity.

Acceleration equals the net force divided by the mass. If the net force acting on an object is doubled, the object’s acceleration will be doubled. Suppose instead that the mass is doubled. Then the acceleration will be halved. If both the net force and the mass are doubled, then the acceleration will be unchanged.

Galileo’s Concept of Inertia | Describing Motion - Aristotle’s ideas were accepted as fact for nearly 2,000 years. Then, in the early 1500s, the Italian scientist Galileo demolished Aristotle’s belief that heavy objects fall faster than light ones. According to legend, Galileo dropped both heavy and light objects from the Leaning Tower of Pisa (Figure 1). He showed that, except for the effects of air friction, objects of different weights fell to the ground in the same amount of time.

Galileo made another discovery. He showed that Aristotle was wrong about forces being necessary to keep objects in motion. Although a force is needed to start an object moving, Galileo showed that, once it is moving, no force is needed to keep it moving except for the force needed to overcome friction. When friction is absent, a moving object needs no force to keep it moving. It will remain in motion all by it self. Rather than philosophizing about ideas, Galileo did something that was quite remarkable at the time. 

Galileo tested his revolutionary idea by  experiment. This was the beginning of modern science. He rolled balls down inclined planes and observed and recorded the gain in speed as rolling continued (Figure 2). On downward-sloping planes, the force of gravity increases a ball’s speed. On an upward slope, the force of gravity decreases a ball’s speed. What about a ball rolling on a level surface? While rolling on a level surface, the ball neither rolls with nor against the vertical force of gravity it neither speeds up nor slows down. The rolling ball maintains a constant speed. Galileo reasoned that a ball moving horizontally would move forever, if friction were entirely absent (Figure 3). Such a ball would move all by itself of its own inertia.

Aristotle on Motion | Describing Motion | Physics - Some two thousand years ago, Greek scientists understood some of the physics we understand today. They had a good grasp of the physics of floating objects and of some of the properties of light, but they were confused about motion.

One of the first to study motion seriously was Aristotle, the most outstanding philosopher-scientist in ancient Greece. Aristotle attempted to clarify motion by classification. 

He classified all motion into two kinds of motion: natural motion and violent motion. We shall briefly consider each, not as study material but as a background to modern ideas about motion.

In Aristotle’s view, natural motion proceeds from the “nature” of an object. He believed that all objects were some combination of four elements—earth, water, air, and fire and he asserted that motion depends on the particular combination of elements an object contains. He taught that every object in the universe has a proper place, which is determined by its “nature”; any object not in its proper place will “strive” to get there. For example, an unsupported lump of clay, being of the earth, properly falls to the ground; an unimpeded puff of smoke, being of the air, properly rises; a feather properly falls to the ground, but not as rapidly as a lump of clay, because it is a mixture of air and earth. Aristotle stated that heavier objects would strive harder and fall faster than lighter ones.

Natural motion was understood to be either straight up or straight down, as in the case of all things on Earth. Natural motion beyond Earth, such as the motion of celestial objects, was circular. Both the Sun and Moon continually circle the Earth in paths without beginning or end. Aristotle taught thatdifferent rules apply in the heavens and that celestial bodies are perfect spheres made of a perfect and unchanging substance, which he called quintessence.*

Violent motion, Aristotle’s other class of motion, is produced by pushes and pulls. Violent motion is imposed motion. A person pushing a cart or lifting a heavy boulder imposes motion, as does someone hurling a stone or winning a tug-of-war. The wind imposes motion on ships. Floodwaters impose it on boulders and tree trunks. Violent motion is externally caused and is imparted to objects, which move not of themselves, not by their nature, but because of impressed forcespushes or pulls.