Bernoulli's principle, sometimes known as Bernoulli's equation, holds that for fluids in an ideal state, pressure and density are inversely related: in other words, a slow-moving fluid exerts more pressure than a fast-moving fluid. Since "fluid" in this context applies equally to liquids and gases, the principle has as many applications with regard to airflow as to the flow of liquids. One of the most dramatic everyday examples of Bernoulli's principle can be found in the airplane, which stays aloft due to pressure differences on the surface of its wing; but the truth of the principle is also illustrated in something as mundane as a shower curtain that billows inward.
HOW IT WORKS
The Swiss mathematician and physicist Daniel Bernoulli (1700-1782) discovered the principle that bears his name while conducting experiments concerning an even more fundamental concept: the conservation of energy. This is a law of physics that holds that a system isolated from all outside factors maintains the same total amount of energy, though energy transformations from one form to another take place.
For instance, if you were standing at the top of a building holding a baseball over the side, the ball would have a certain quantity of potential energy—the energy that an object possesses by virtue of its position. Once the ball is dropped, it immediately begins losing potential energy and gaining kinetic energy—the energy that an object possesses by virtue of its motion. Since the total energy must remain constant, potential and kinetic energy have an inverse relationship: as the value of one variable decreases, that of the other increases in exact proportion.
The ball cannot keep falling forever, losing potential energy and gaining kinetic energy. In fact, it can never gain an amount of kinetic energy greater than the potential energy it possessed in the first place. At the moment before the ball hits the ground, its kinetic energy is equal to the potential energy it possessed at the top of the building. Correspondingly, its potential energy is zero—the same amount of kinetic energy it possessed before it was dropped.
Then, as the ball hits the ground, the energy is dispersed. Most of it goes into the ground, and depending on the rigidity of the ball and the ground, this energy may cause the ball to bounce. Some of the energy may appear in the form of sound, produced as the ball hits bottom, and some will manifest as heat. The total energy, however, will not be lost: it will simply have changed form.
Bernoulli was one of the first scientists to propose what is known as the kinetic theory of gases: that gas, like all matter, is composed of tiny molecules in constant motion. In the 1730s, he conducted experiments in the conservation of energy using liquids, observing how water flows through pipes of varying diameter. In a segment of pipe with a relatively large diameter, he observed, water flowed slowly, but as it entered a segment of smaller diameter, its speed increased.
It was clear that some force had to be acting on the water to increase its speed. Earlier, Robert Boyle (1627-1691) had demonstrated that pressure and volume have an inverse relationship, and Bernoulli seems to have applied Boyle's findings to the present situation. Clearly the volume of water flowing through the narrower pipe at any given moment was less than that flowing through the wider one. This suggested, according to Boyle's law, that the pressure in the wider pipe must be greater.
As fluid moves from a wider pipe to a narrower one, the volume of that fluid that moves a given distance in a given time period does not change. But since the width of the narrower pipe is smaller, the fluid must move faster in order to achieve that result. One way to illustrate this is to observe the behavior of a river: in a wide, unconstricted region, it flows slowly, but if its flow is narrowed by canyon walls (for instance), then it speeds up dramatically.
The above is a result of the fact that water is a fluid, and having the characteristics of a fluid, it adjusts its shape to fit that of its container or other solid objects it encounters on its path. Since the volume passing through a given length of pipe during a given period of time will be the same, there must be a decrease in pressure. Hence Bernoulli's conclusion: the slower the rate of flow, the higher the pressure, and the faster the rate of flow, the lower the pressure.
Bernoulli published the results of his work in Hydrodynamica (1738), but did not present his ideas or their implications clearly. Later, his friend the German mathematician Leonhard Euler (1707-1783) generalized his findings in the statement known today as Bernoulli's principle.
The Venturi Tube
Also significant was the work of the Italian physicist Giovanni Venturi (1746-1822), who is credited with developing the Venturi tube, an instrument for measuring the drop in pressure that takes place as the velocity of a fluid increases. It consists of a glass tube with an inward-sloping area in the middle, and manometers, devices for measuring pressure, at three places: the entrance, the point of constriction, and the exit. The Venturi meter provided a consistent means of demonstrating Bernoulli's principle.
Like many propositions in physics, Bernoulli's principle describes an ideal situation in the absence of other forces. One such force is viscosity, the internal friction in a fluid that makes it resistant to flow. In 1904, the German physicist Ludwig Prandtl (1875-1953) was conducting experiments in liquid flow, the first effort in well over a century to advance the findings of Bernoulli and others. Observing the flow of liquid in a tube, Prandtl found that a tiny portion of the liquid adheres to the surface of the tube in the form of a thin film, and does not continue to move. This he called the viscous boundary layer.
Like Bernoulli's principle itself, Prandtl's findings would play a significant part in aerodynamics, or the study of airflow and its principles. They are also significant in hydrodynamics, or the study of water flow and its principles, a discipline Bernoulli founded.
Laminar vs. Turbulent Flow
Air and water are both examples of fluids, substances which—whether gas or liquid—conform to the shape of their container. The flow patterns of all fluids may be described in terms either of laminar flow, or of its opposite, turbulent flow.
Laminar flow is smooth and regular, always moving at the same speed and in the same direction. Also known as streamlined flow, it is characterized by a situation in which every particle of fluid that passes a particular point follows a path identical to all particles that passed that point earlier. A good illustration of laminar flow is what occurs when a stream flows around a twig.
By contrast, in turbulent flow, the fluid is subject to continual changes in speed and direction—as, for instance, when a stream flows over shoals of rocks. Whereas the mathematical model of laminar flow is rather straightforward, conditions are much more complex in turbulent flow, which typically occurs in the presence of obstacles or high speeds.
Turbulent flow makes it more difficult for two streams of air, separated after hitting a barrier, to rejoin on the other side of the barrier; yet that is their natural tendency. In fact, if a single air current hits an airfoil—the design of an airplane's wing when seen from the end, a streamlined shape intended to maximize the aircraft's response to airflow—the air that flows over the top will "try" to reach the back end of the airfoil at the same time as the air that flows over the bottom. In order to do so, it will need to speed up—and this, as will be shown below, is the basis for what makes an airplane fly.
When viscosity is absent, conditions of perfect laminar flow exist: an object behaves in complete alignment with Bernoulli's principle. Of course, though ideal conditions seldom occur in the real world, Bernoulli's principle provides a guide for the behavior of planes in flight, as well as a host of everyday things.
For thousands of years, human beings vainly sought to fly "like a bird," not realizing that this is literally impossible, due to differences in physiognomy between birds and homo sapiens. No man has ever been born (or ever will be) who possesses enough strength in his chest that he could flap a set of attached wings and lift his body off the ground. Yet the bird's physical structure proved highly useful to designers of practical flying machines.
A bird's wing is curved along the top, so that when air passes over the wing and divides, the curve forces the air on top to travel a greater distance than the air on the bottom. The tendency of airflow, as noted earlier, is to correct for the presence of solid objects and to return to its original pattern as quickly as possible. Hence, when the air hits the front of the wing, the rate of flow at the top increases to compensate for the greater distance it has to travel than the air below the wing. And as shown by Bernoulli, fast-moving fluid exerts less pressure than slow-moving fluid; therefore, there is a difference in pressure between the air below and the air above, and this keeps the wing aloft.
Only in 1853 did Sir George Cayley (1773-1857) incorporate the avian airfoil to create history's first workable (though engine-less) flying machine, a glider. Much, much older than Cayley's glider, however, was the first manmade flying machine built "according to Bernoulli's principle"—only it first appeared in about 12,000
b.c., and the people who created it had had little contact with the outside world until the late eighteenth century a.d. This was the boomerang, one of the most ingenious devices ever created by a stone-age society—in this case, the Aborigines of Australia.
Contrary to the popular image, a boomerang flies through the air on a plane perpendicular to the ground, rather than parallel. Hence, any thrower who properly knows how tosses the boomerang not with a side-arm throw, but overhand. As it flies, the boomerang becomes both a gyroscope and an airfoil, and this dual role gives it aerodynamic lift.
Like the gyroscope, the boomerang imitates a top; spinning keeps it stable. It spins through the air, its leading wing (the forward or upward wing) creating more lift than the other wing. As an airfoil, the boomerang is designed so that the air below exerts more pressure than the air above, which keeps it airborne.
Another very early example of a flying machine using Bernoulli's principles is the kite, which first appeared in China in about 1000 b.c. The kite's design, particularly its use of lightweight fabric stretched over two crossed strips of very light wood, makes it well-suited for flight, but what keeps it in the air is a difference in air pressure. At the best possible angle of attack, the kite experiences an ideal ratio of pressure from the slower-moving air below versus the faster-moving air above, and this gives it lift.
Later Cayley studied the operation of the kite, and recognized that it—rather than the balloon, which at first seemed the most promising apparatus for flight—was an appropriate model for the type of heavier-than-air flying machine he intended to build. Due to the lack of a motor, however, Cayley's prototypical airplane could never be more than a glider: a steam engine, then state-of-the-art technology, would have been much too heavy.
Hence, it was only with the invention of the internal-combustion engine that the modern airplane came into being. On December 17, 1903, at Kitty Hawk, North Carolina, Orville (1871-1948) and Wilbur (1867-1912) Wright tested a craft that used a 25-horsepower engine they had developed at their bicycle shop in Ohio. By maximizing the ratio of power to weight, the engine helped them overcome the obstacles that had dogged recent attempts at flight, and by the time the day was over, they had achieved a dream that had eluded men for more than four millennia.
Within fifty years, airplanes would increasingly obtain their power from jet rather than internal-combustion engines. But the principle that gave them flight, and the principle that kept them aloft once they were airborne, reflected back to Bernoulli's findings of more than 160 years before their time. This is the concept of the airfoil.
As noted earlier, an airfoil has a streamlined design. Its shape is rather like that of an elongated, asymmetrical teardrop lying on its side, with the large end toward the direction of airflow, and the narrow tip pointing toward the rear. The greater curvature of its upper surface in comparison to the lower side is referred to as the airplane's camber. The front end of the airfoil is also curved, and the chord line is an imaginary straight line connecting the spot where the air hits the front—known as the stagnation point—to the rear, or trailing edge, of the wing.
Again, in accordance with Bernoulli's principle, the shape of the airflow facilitates the spread of laminar flow around it. The slower-moving currents beneath the airfoil exert greater pressure than the faster currents above it, giving lift to the aircraft. Of course, the aircraft has to be moving at speeds sufficient to gain momentum for its leap from the ground into the air, and here again, Bernoulli's principle plays a part.
Thrust comes from the engines, which run the propellers—whose blades in turn are designed as miniature airfoils to maximize their power by harnessing airflow. Like the aircraft wings, the blades' angle of attack—the angle at which airflow hits it. In stable flight, the pilot greatly increases the angle of attack (also called pitched), whereas at takeoff and landing, the pitch is dramatically reduced.
Drawing Fluids Upward: Atomizers and Chimneys
A number of everyday objects use Bernoulli's principle to draw fluids upward, and though in terms of their purposes, they might seem very different—for instance, a perfume atomizer vs. a chimney—they are closely related in their application of pressure differences. In fact, the idea behind an atomizer for a perfume spray bottle can also be found in certain garden-hose attachments, such as those used to provide a high-pres-sure car wash.
The air inside the perfume bottle is moving relatively slowly; therefore, according to Bernoulli's principle, its pressure is relatively high, and it exerts a strong downward force on the perfume itself. In an atomizer there is a narrow tube running from near the bottom of the bottle to the top. At the top of the perfume bottle, it opens inside another tube, this one perpendicular to the first tube. At one end of the horizontal tube is a simple squeeze-pump which causes air to flow quickly through it. As a result, the pressure toward the top of the bottle is reduced, and the perfume flows upward along the vertical tube, drawn from the area of higher pressure at the bottom. Once it is in the upper tube, the squeeze-pump helps to eject it from the spray nozzle.
A carburetor works on a similar principle, though in that case the lower pressure at the top draws air rather than liquid. Likewise a chimney draws air upward, and this explains why a windy day outside makes for a better fire inside. With wind blowing over the top of the chimney, the air pressure at the top is reduced, and tends to draw higher-pressure air from down below.
The upward pull of air according to the Bernoulli principle can also be illustrated by what is sometimes called the "Hoover bugle"—a name perhaps dating from the Great Depression, when anything cheap or contrived bore the appellation "Hoover" as a reflection of popular dissatisfaction with President Herbert Hoover. In any case, the Hoover bugle is simply a long corrugated tube that, when swung overhead, produces musical notes.
You can create a Hoover bugle using any sort of corrugated tube, such as vacuum-cleaner hose or swimming-pool drain hose, about 1.8 in (4 cm) in diameter and 6 ft (1.8 m) in length. To operate it, you should simply hold the tube in both hands, with extra length in the leading hand—that is, the right hand, for most people. This is the hand with which to swing the tube over your head, first slowly and then faster, observing the changes in tone that occur as you change the pace.
The vacuum hose of a Hoover tube can also be returned to a version of its original purpose in an illustration of Bernoulli's principle. If a piece of paper is torn into pieces and placed on a table, with one end of the tube just above the paper and the other end spinning in the air, the paper tends to rise. It is drawn upward as though by a vacuum cleaner—but in fact, what makes it happen is the pressure difference created by the movement of air.
In both cases, reduced pressure draws air from the slow-moving region at the bottom of the tube. In the case of the Hoover bugle, the corrugations produce oscillations of a certain frequency. Slower speeds result in slower oscillations and hence lower frequency, which produces a lower tone. At higher speeds, the opposite is true. There is little variation in tones on a Hoover bugle: increasing the velocity results in a frequency twice that of the original, but it is difficult to create enough speed to generate a third tone.
Spin, Curve, and Pull: The Counterintuitive Principle
There are several other interesting illustrations—sometimes fun and in one case potentially tragic—of Bernoulli's principle. For instance, there is the reason why a shower curtain billows inward once the shower is turned on. It would seem logical at first that the pressure created by the water would push the curtain outward, securing it to the side of the bathtub.
Instead, of course, the fast-moving air generated by the flow of water from the shower creates a center of lower pressure, and this causes the curtain to move away from the slower-moving air outside. This is just one example of the ways in which Bernoulli's principle creates results that, on first glance at least, seem counterintuitive—that is, the opposite of what common sense would dictate.
Another fascinating illustration involves placing two empty soft drink cans parallel to one another on a table, with a couple of inches or a few centimeters between them. At that point, the air on all sides has the same slow speed. If you were to blow directly between the cans, however, this would create an area of low pressure between them. As a result, the cans push together. For ships in a harbor, this can be a frightening prospect: hence, if two crafts are parallel to one another and a strong wind blows between them, there is a possibility that they may behave like the cans.
Then there is one of the most illusory uses of Bernoulli's principle, that infamous baseball pitcher's trick called the curve ball. As the ball moves through the air toward the plate, its velocity creates an air stream moving against the trajectory of the ball itself. Imagine it as two lines, one curving over the ball and one curving under, as the ball moves in the opposite direction.
In an ordinary throw, the effects of the airflow would not be particularly intriguing, but in this case, the pitcher has deliberately placed a "spin" on the ball by the manner in which he has thrown it. How pitchers actually produce spin is a complex subject unto itself, involving grip, wrist movement, and other factors, and in any case, the fact of the spin is more important than the way in which it was achieved.
If the direction of airflow is from right to left, the ball, as it moves into the airflow, is spinning clockwise. This means that the air flowing over the ball is moving in a direction opposite to the spin, whereas that flowing under it is moving in the same direction. The opposite forces produce a drag on the top of the ball, and this cuts down on the velocity at the top compared to that at the bottom of the ball, where spin and airflow are moving in the same direction.
Thus the air pressure is higher at the top of the ball, and as per Bernoulli's principle, this tends to pull the ball downward. The curve ball—of which there are numerous variations, such as the fade and the slider—creates an unpredictable situation for the batter, who sees the ball leave the pitcher's hand at one altitude, but finds to his dismay that it has dropped dramatically by the time it crosses the plate.
A final illustration of Bernoulli's often counterintuitive principle neatly sums up its effects on the behavior of objects. To perform the experiment, you need only an index card and a flat surface. The index card should be folded at the ends so that when the card is parallel to the surface, the ends are perpendicular to it. These folds should be placed about half an inch (about one centimeter) from the ends.
At this point, it would be handy to have an unsuspecting person—someone who has not studied Bernoulli's principle—on the scene, and challenge him or her to raise the card by blowing under it. Nothing could seem easier, of course: by blowing under the card, any person would naturally assume, the air will lift it. But of course this is completely wrong according to Bernoulli's principle. Blowing under the card, as illustrated, will create an area of high velocity and low pressure. This will do nothing to lift the card: in fact, it only pushes the card more firmly down on the table.
WHERE TO LEARN MORE
Beiser, Arthur. Physics, 5th ed. Reading, MA: Addison-Wesley, 1991.
"Bernoulli's Principle: Explanations and Demos." (Web site). <http://126.96.36.199/physicszone/lesson/02forces/bernoull/bernoull.html> (February 22, 2001).
Cockpit Physics (Department of Physics, United States Air Force Academy. Web site.). <http://www.usafa.af.mil/dfp/cockpit-phys/> (February 19, 2001).
K8AIT Principles of Aeronautics Advanced Text. (Web site). <http://wings.ucdavis.edu/Book/advanced.html> (February 19, 2001).
Schrier, Eric and William F. Allman. Newton at the Bat: The Science in Sports. New York: Charles Scribner's Sons, 1984.
Smith, H. C. The Illustrated Guide to Aerodynamics. Blue Ridge Summit, PA: Tab Books, 1992.
Stever, H. Guyford, James J. Haggerty, and the Editors of Time-Life Books. Flight. New York: Time-Life Books, 1965.
The study of airflow and its principles. Applied aerodynamics is the science of improving man-made objects in light of those principles.
The design of an airplane's wing when seen from the end, a shape intended to maximize the aircraft's response to airflow.
ANGLE OF ATTACK:
The orientation of the airfoil with regard to the airflow, or the angle that the chord line forms with the direction of the air stream.
A proposition, credited to Swiss mathematician and physicist Daniel Bernoulli (1700-1782), which maintains that slower-moving fluid exerts greater pressure than faster-movingfluid.
The enhanced curvature on the upper surface of an airfoil.
The distance, along an imaginary straight line, from the stagnation point of an airfoil to the rear, or trailing edge.
CONSERVATION OF ENERGY:
A law of physics which holds that within a system isolated from all other outside factors, the total amount of energy remains the same, though transformations of energy from one form to another take place.
Any substance, whether gas or liquid, that conforms to the shape of itscontainer.
The study of water flow and its principles.
A situation involving two variables, in which one of the two increases in direct proportion to the decrease in the other.
The energy that an object possesses by virtue of its motion.
A term describing a streamlined flow, in which all particles move at the same speed and in the same direction. Its opposite is turbulent flow.
An aerodynamic force perpendicular to the direction of the wind. For an aircraft, lift is the force that raises it off the ground and keeps it aloft.
Devices for measuring pressure in conjunction with a Venturi tube.
The energy that an object possesses by virtue of its position.
The spot where airflow hits the leading edge of an airfoil.
A term describing a highly irregular form of flow, in which a fluid is subject to continual changes in speed and direction. Its opposite is laminar flow.
An instrument, consisting of a glass tube with an inward-sloping area in the middle, for measuring the drop in pressure that takes place as the velocity of a fluid increases.
The internal friction in a fluid that makes it resistant to flow.
Bernoulli's principle states that flowing fluids like air and water press less than still fluids and that pressure decreases quadratically with speed; i.e., with speed squared.
One quarter of a millennium ago, Daniel Bernoulli pioneered the use of kinetic theory that molecules moved and bumped things. He also knew that flowing fluids pressed less, but he did not connect these ideas logically. In Hydrodynamica, Daniel's logic that flow reduced pressure was obscure, and his formula was awkward. Daniel's father Johann, amid controversy, improved his son's insight and presentation in Hydraulica. This research was centered in St. Petersburg where Leonhard Euler, a colleague of Daniel and a student of Johann, generalized a rate-of-change dependence of pressure and density on speed of flow. Bernoulli's principle for liquids was then formulated in modern form for the first time.
In this same group of scientists was d'Alembert, who found paradoxically that fluids stopped ahead of obstacles, so frictionless flow did not push.
Progress then seems to have halted for about a century and a half until Ludwig Prandtl or one of his students solved Euler's equation for smooth streams of air in order to have a mathematical model of flowing air for designing wings. Here, speed lowers pressure more than it lowers density because expanding air cools, and the ratio of density times degrees-kelvin divided by pressure is constant for an ideal gas.
More turbulent flow, as in atmospheric winds, requires an alternative solution of Euler's equation because mixing keeps air-temperature fixed.
Applying Bernoulli's principle
Bernoulli's principle is regarded by many as a paradox because currents and winds upset things, but standing a stick in a stream of water helps to clarify the enigma. You will see calm, smooth, level water ahead of the stick and a cavity of reduced pressure behind it. Calm water pushes the stick, as lower pressure downstream fails to balance the upsetting force .
Bernoulli's principle never acts alone; it also comes with molecular entrainment. Molecules in the lower pressure of faster flow aspirate and whisk away molecules from the higher pressure of slower flow. Solid obstacles such as airfoils carry a very thin stagnant layer of air with them. A swift low-pressure airstream takes some molecules from this boundary layer and reduces molecular impacts on that surface of the wing across which the airstream moves faster.
For Bernoulli's principle to dominate a dynamic situation, friction must be less dominant. Elastic molecular impacts are frictionless—no heating. Molecules of dry air, even more than those of water, collide elastically; so Bernoulli's principle with its molecular-entrainment agent is dominate for windy air.
See also Aerodynamics.
Bernoulli's principle describes the relationship between the pressure and the velocity of a moving fluid (i.e., air or water ). Bernoulli's principle states that as the velocity of fluid flow increases, the pressure exerted by that fluid decreases.
During the late eighteenth century, Daniel Bernoulli pioneered the basic tenet of kinetic theory, that molecules are in motion. He also knew that flowing fluids exerted less pressure, but he did not connect these ideas logically. In Hydrodynamica, Bernoulli's logic that flow reduced pressure was obscure, and his formula was awkward. Bernoulli's father Johann, amid controversy, improved his son's insight and presentation in Hydraulica. This research was centered in St. Petersburg where Leonard Euler, a colleague of Bernoulli and a student of Johann, generalized a rate-of-change dependence of pressure and density on speed of flow. Bernoulli's principle for liquids was then formulated in modern form for the first time.
In this same group of scientists was D'Alembert, who found paradoxically that fluids stopped ahead of obstacles, so frictionless flow did not push.
Progress then seems to have halted for about a century and a half until Ludwig Prandtl or one of his students solved Euler's equation for smooth streams of air in order to have a mathematical model of flowing air for designing wings. Here, speed lowers pressure more than it lowers density because expanding air cools, and the ratio of density times degrees-kelvin divided by pressure is constant for an ideal-gas.
More turbulent flow, as in atmospheric winds, requires an alternative solution of Euler's equation because mixing keeps air-temperature fixed.
Bernoulli's principle is regarded by many as a paradox because currents and winds upset things, but standing a stick in a stream of water helps to clarify the enigma. One can observe calm, smooth, level water ahead of the stick and a cavity of reduced pressure behind it. Calm water pushes the stick, as lower pressure downstream fails to balance the upsetting force.
Bernoulli's principle never acts alone; it also comes with molecular entrainment. Molecules in the lower pressure of faster flow aspirate and whisk away molecules from the higher pressure of slower flow. Solid obstacles such as airfoils carry a thin stagnant layer of air with them. A swift low-pressure airstream takes some molecules from this boundary layer and reduces molecular impacts on that surface of the wing across which the airstream moves faster.
See also Atmospheric chemistry; Hydrostatic pressure
Bernoulli's principle, physical principle formulated by Daniel Bernoulli that states that as the speed of a moving fluid (liquid or gas) increases, the pressure within the fluid decreases. The phenomenon described by Bernoulli's principle has many practical applications; it is employed in the carburetor and the atomizer, in which air is the moving fluid, and in the aspirator, in which water is the moving fluid. In the first two devices air moving through a tube passes through a constriction, which causes an increase in speed and a corresponding reduction in pressure. As a result, liquid is forced up into the air stream (through a narrow tube that leads from the body of the liquid to the constriction) by the greater atmospheric pressure on the surface of the liquid. In the aspirator air is drawn into a stream of water as the water flows through a constriction. Bernoulli's principle can be explained in terms of the law of conservation of energy (see conservation laws, in physics). As a fluid moves from a wider pipe into a narrower pipe or a constriction, a corresponding volume must move a greater distance forward in the narrower pipe and thus have a greater speed. At the same time, the work done by corresponding volumes in the wider and narrower pipes will be expressed by the product of the pressure and the volume. Since the speed is greater in the narrower pipe, the kinetic energy of that volume is greater. Then, by the law of conservation of energy, this increase in kinetic energy must be balanced by a decrease in the pressure-volume product, or, since the volumes are equal, by a decrease in pressure.