Chapter 15. Human Movement in a Fluid Medium

After completing this chapter, you will be able to:

• Explain the ways in which the composition and flow characteristics of a fluid affect fluid forces.
• Define buoyancy and explain the variables that determine whether a human body will float.
• Define drag, identify the components of drag, and identify the factors that affect the magnitude of each component.
• Define lift and explain the ways in which it can be generated.
• Discuss the theories regarding propulsion of the human body in swimming.

Why are there dimples in a golf ball? Why are some people able to float while others cannot? Why are cyclists, swimmers, downhill skiers, and speed skaters concerned with streamlining their bodies during competition?

Both air and water are fluid mediums that exert forces on bodies moving through them. Some of these forces slow the progress of a moving body; others provide support or propulsion. A general understanding of the actions of fluid forces on human movement activities is an important component of the study of the biomechanics of human movement. This chapter introduces the effects of fluid forces on both human and projectile motion.

Although in general conversation the term fluid is often used interchangeably with the term liquid, from a mechanical perspective, a fluid is any substance that tends to flow or continuously deform when acted on by a shearforce (42). Both gases and liquids are fluids with similar mechanical behaviors.

Relative Motion

Because a fluid is a medium capable of flow, the influence of the fluid on a body moving through it depends not only on the body’s velocity but also on the velocity of the fluid. Consider the case of waders standing in the shallow portion of a river with a moderately strong current. If they stand still, they feel the force of the current against their legs. If they walk upstream against the current, the current’s force against their legs is even stronger. If they walk downstream, the current’s force is reduced and perhaps even imperceptible.

When a body moves through a fluid, the relative velocity of the body with respect to the fluid influences the magnitude of the acting forces. If the direction of motion is directly opposite the direction of the fluid flow, the magnitude of the velocity of the moving body relative to the fluid is the algebraic sum of the speeds of the moving body and the fluid (Figure 15-1). If the body moves in the same direction as the surrounding fluid, the magnitude of the body’s velocity relative to the fluid is the difference in the speeds of the object and the fluid. In other words, the relative velocity of a body with respect to a fluid is the vector subtraction of the absolute velocity of the fluid from the absolute velocity of the body (see Sample Problem 15.1). Likewise, the relative velocity of a fluid with respect to a body moving through it is the vector subtraction of the velocity of the body from the velocity of the fluid.

Sample Problem 15.1

A sailboat is traveling at an absolute speed of 3 m/s against a 0.5 m/s current and with a 6 m/s tailwind. What is the velocity of the current with respect to the boat? What is the velocity of the wind with respect to the boat?

Known

Solution

The velocity of the current with respect to the boat is equal to the vector subtraction of the absolute velocity of the boat from the absolute velocity of the current.

The velocity of the wind with respect to the boat is equal to the vector subtraction of the absolute velocity of the boat from the absolute velocity of the wind.

Laminar versus Turbulent Flow

When an object such as a human hand or a canoe paddle moves through water, there is little apparent disturbance of the immediately surrounding water if the relative velocity of the object with respect to the water is low. However, if the relative velocity of motion through the water is sufficiently high, waves and eddies appear.

When an object moves with sufficiently low velocity relative to any fluid medium, the flow of the adjacent fluid is termed laminar flow. Laminar flow is characterized by smooth layers of fluid molecules flowing parallel to one another (Figure 15-2).

When an object moves with sufficiently high velocity relative to a surrounding fluid, the layers of fluid near the surface of the object mix, and the flow is termed turbulent. The rougher the surface of the body, the lower the relative velocity at which turbulence is caused. Laminar flow and turbulent flow are distinct categories. If any turbulence is present, the flow is nonlaminar. The nature of the fluid flow surrounding an object can dramatically affect the fluid forces exerted on the object. In the case of the human body during swimming, flow is neither completely laminar nor completely turbulent, but transitional between the two (34).

Fluid Properties

Other factors that influence the magnitude of the forces a fluid generates are the fluid’s density, specific weight, and viscosity. As discussed in Chapter 3, density (ρ) is defined as mass/volume, and the ratio of weight to volume is known as specific weight (γ). The denser and heavier the fluid medium surrounding a body, the greater the magnitude of the forces the fluid exerts on the body. The property of fluid viscosity involves the internal resistance of a fluid to flow. The greater the extent to which a fluid resists flow under an applied force, the more viscous the fluid is. A thick molasses, for example, is more viscous than a liquid honey, which is more viscous than water. Increased fluid viscosity results in increased forces exerted on bodies exposed to the fluid.

Atmospheric pressure and temperature influence a fluid’s density, specific weight, and viscosity, with more mass concentrated in a given unit of fluidvolume at higher atmospheric pressures and lower temperatures. Because molecular motion in gases increases with temperature, the viscosity of gases also increases. The viscosity of liquids decreases with increased temperature because of a reduction in the cohesive forces among the molecules. The densities, specific weights, and viscosities of common fluids are shown in Table 15-1.

Characteristics of the Buoyant Force

Buoyancy is a fluidforce that always acts vertically upward. The factors that determine the magnitude of the buoyant force were originally explained by the ancient Greek mathematician Archimedes. Archimedes’ principle states that the magnitude of the buoyant force acting on a given body is equal to the weight of the fluid displaced by the body. The latter factor is calculated by multiplying the specific weight of the fluid by the volume of the portion of the body that is surrounded by the fluid. Buoyancy (Fb) is calculated as the product of the displaced volume (Vd) and the fluid’s specific weight γ:

For example, if a water polo ball with a volume of 0.2 m3 is completely submerged in water at 20˚C, the buoyant force acting on the ball is equal to the ball’s volume multiplied by the specific weight of water at 20˚C:

The more dense the surrounding fluid, the greater the magnitude of the buoyant force. Since seawater is more dense than fresh water, a given object’s buoyancy is greater in seawater than in fresh water. Because the magnitude of the buoyant force is directly related to the volume of the submerged object, the point at which the buoyant force acts is the object’s center of volume, which is also known as the center of buoyancy. The center of volume is the point around which a body’s volume is equally distributed.

Flotation

The ability of a body to float in a fluid medium depends on the relationship between the body’s buoyancy and its weight. When weight and the buoyant force are the only two forces acting on a body and their magnitudes are equal, the body floats in a motionless state, in accordance with the principles of static equilibrium. If the magnitude of the weight is greater than that of the buoyant force, the body sinks, moving downward in the direction of the net force.

Most objects float statically in a partially submerged position. The volume of a freely floating object needed to generate a buoyant force equal to the object’s weight is the volume that is submerged.

Flotation of the Human Body

In the study of biomechanics, buoyancy is most commonly of interest relative to the flotation of the human body in water. Some individuals cannot float in a motionless position, and others float with little effort. This difference in floatability is a function of body density. Since the density of bone and muscle is greater than the density of fat, individuals who are extremely muscular and have little body fat have higher average body densities than individuals with less muscle, less dense bones, or more body fat. If two individuals have an identical body volume, the one with the higher body density weighs more. Alternatively, if two people have the same body weight, the person with the higher body density has a smaller body volume. For flotation to occur, the body volume must be large enough to create a buoyant force greater than or equal to body weight (see Sample Problem 15.2). Many individuals can float only when holding a large volume of inspired air in the lungs, a tactic that increases body volume without altering body weight.

The orientation of the human body as it floats in water is determined by the relative position of the total-body center of gravity relative to the total-body center of volume. The exact locations of the center of gravity and center of volume vary with anthropometric dimensions and body composition. Typically, the center of gravity is inferior to the center of volume due to the relatively large volume and relatively small weight of the lungs. Because weight acts at the center of gravity and buoyancy acts at the center of volume, a torque is created that rotates the body until it is positioned so that these two acting forces are vertically aligned and the torque ceases to exist (Figure 15-3).

When beginning swimmers try to float on their backs, they typically assume a horizontal body position. Once the swimmer relaxes, the lower end of the body sinks, because of the acting torque. An experienced teacher instructs beginning swimmers to assume a more diagonal position in the water before relaxing into the back float. This position minimizes torque and the concomitant sinking of the lower extremity. Other strategies that a swimmer can use to reduce torque on the body when entering a back float position include extending the arms backward in the water above the head and flexing the knees. Both tactics elevate the location of the center of gravity, positioning it closer to the center of volume.

During swimming with the front crawl stroke, the center of buoyancy is shifted toward the feet when the recovery arm and part of the head are above the surface of the water. At this point in the stroke cycle, the buoyant torque tends to elevate the feet, rather than the reverse (55).

Sample Problem 15.2

When holding a large quantity of inspired air in her lungs, a 22 kg girl has a body volume of 0.025 m3. Can she float in fresh water if γ equals 9810 N/m3? Given her body volume, how much could she weigh and still be able to float?

Known

Solution

Two forces are acting on the girl: her weight and the buoyant force. According to the conditions of static equilibrium, the sum of the vertical forces must be equal to zero for the girl to float in a motionless position. If the buoyant force is less than her weight, she will sink, and if the buoyant force is equal to her weight, she will float completely submerged. If the buoyant force is greater than her weight, she will float partly submerged. The magnitude of the buoyant force acting on her total body volume is the product of the volume of displaced fluid (her body volume) and the specific weight of the fluid:

Her body weight is equal to her body mass multiplied by the acceleration of gravity:

Since the buoyant force is greater than her body weight, the girl will float partly submerged in fresh water.

To calculate the maximum weight that the girl’s body volume can support in fresh water, multiply the body volume by the specific weight of water.

Drag is a force caused by the dynamic action of a fluid that acts in the direction of the free-stream fluid flow. Generally, a drag is a resistance force: a force that slows the motion of a body moving through a fluid. The drag force acting on a body in relative motion with respect to a fluid is defined by the following formula:

In this formula, FD is drag force, CD is the coefficient of drag, ρ is the fluiddensity, Ap is the projected area of the body or the surface area of the body oriented perpendicular to the fluid flow, and v is the relative velocity of the body with respect to the fluid. The coefficient of drag is a unitless number that serves as an index of the amount of drag an object can generate. Its size depends on the shape and orientation of a body relative to the fluid flow, with long, streamlined bodies generally having lower coefficients of drag than blunt or irregularly shaped objects. Approximate coefficients of drag for the human body in positions commonly assumed during participation in several sports are shown in Figure 15-4.

The formula for the total drag force demonstrates the exact way in which each of the identified factors affects drag. If the coefficient of drag, the fluiddensity, and the projected area of the body remain constant, drag increases with the square of the relative velocity of motion. This relationship is referred to as the theoretical square law. According to this law, if cyclists double their speed and other factors remain constant, the drag force opposing them increases fourfold. The effect of drag is more consequential when a body is moving with a high velocity, which occurs in sports such as cycling, speed skating, downhill skiing, the bobsled, and the luge.

Increase or decrease in the fluiddensity also results in a proportional change in the drag force. Because air density decreases with increasing altitude, many world records set at the 1968 Olympic Games in Mexico City, where the elevation is 2250 m, may have been partially attributable to the reduced air resistance acting on the competitors. Mathematical model–based estimates indicate that the reduction in drag attributable to the lesser air density in Mexico City accounts for 0.08 s of performance time in the 100 m sprint and 0.16 s of race time in the 200 m event (29). Bob Beamon’s noteworthy long jump performance of 8.9 m during the games was 2.4 cm longer than if the same jump had been performed at sea level (53). Theoretical calculations indicate that because air density decreases more than VO2max with increasing altitude, the ideal altitude for cycling performance should be 4000 m (7).

In swimming, the drag on a moving body is 500–600 times higher than it would be in the air, with the magnitude of drag varying with the anthropometric characteristics of the individual swimmer, as well as with the stroke used (46). Researchers distinguish between passive drag, which is generated by the swimmer’s body size, shape, and position in the water, and active drag, which is associated with the swimming motion. Passive drag is inversely related to a swimmer’s buoyancy, which has been found to have a small but important influence on sprint swimming performance (33). Passive drag on male swimmers is also significantly reduced with shoulder-to-knee and shoulder-to-ankle swimsuits as compared to briefs (34). Elite swimmers generate much less active drag than average swimmers through superior stroke technique (27). Older swimmers tend to generate more drag than younger swimmers, and sprinters tend to incur more drag than long-distance swimmers (26). Furthermore, better swimmers appear to be able to increase swimming speed while simultaneously reducing drag or showing only a small increase (26).

Three forms of resistance contribute to the total drag force. The component of resistance that predominates depends on the nature of the fluid flow immediately adjacent to the body.

Skin Friction

One component of the total drag is known as skin friction, surface drag, or viscous drag. This drag is similar to the friction force described in Chapter 12. Skin friction is derived from the sliding contacts between successive layers of fluid close to the surface of a moving body (Figure 15-5). The layer of fluid particles immediately adjacent to the moving body is slowed because of the shear stress the body exerts on the fluid. The next adjacent layer of fluid particles moves with slightly less speed because of friction between the adjacent molecules, and the next layer is affected in turn. The number of layers of affected fluid becomes progressively larger as the flow moves in the downstream direction along the body. The entire region within which fluid velocity is diminished because of the shearing resistance caused by the boundary of the moving body is the boundary layer. The force the body exerts on the fluid in creating the boundary layer results in an oppositely directed reaction force exerted by the fluid on the body. This reaction force is known as skin friction.

Several factors affect the magnitude of skin friction drag. It increases proportionally with increases in the relative velocity of fluid flow, the surface area of the body over which the flow occurs, the roughness of the body surface, and the viscosity of the fluid. Skin friction is always one component of the total drag force acting on a body moving relative to a fluid, and it is the major form of drag present when the flow is primarily laminar. For front crawl swimming, kayaking, and rowing, skin friction drag predominates at velocities between 1 and 3 m/s (38).

Among these factors, the one that a competitive athlete can readily alter is the relative roughness of the body surface. Athletes can wear tight-fitting clothing composed of a smooth fabric rather than loose-fitting clothing or clothing made of a rough fabric. A 10% reduction of drag occurs when a speed skater wears a smooth spandex suit as opposed to the traditional wool outfit (51). A 6% decrease in air resistance results from cyclists’ use of appropriate clothing, including sleeves, tights, and smooth covers over the laces of the shoes (28). The cotton socks and loose shorts and singlets commonly worn by recreational runners are particularly aerodynamically inefficient. Smoothing the running shoe and laces and either covering or shaving body hair reduces drag on runners by as much as 10% (29). Competitive male swimmers and cyclists often shave body hair to reduce skin friction.

The other factor affecting skin friction that athletes can alter in some circumstances is the amount of surface area in contact with the fluid. Carrying an extra passenger such as a coxswain in a rowing event results in a larger wetted surface area of the hull because of the added weight; as a result, skin friction drag is increased.

Form Drag

A second component of the total drag acting on a body moving through a fluid is form drag, which is also known as profile drag or pressure drag. Form drag is always one component of the drag on a body moving relative to a fluid. When the boundary layer of fluid molecules next to the surface of the moving body is primarily turbulent, form drag predominates. Form drag is the major contributor to overall drag during most human and projectile motion. It is the predominant type of drag for front crawl swimming, kayaking, and rowing at velocities of less than 1 m/s (38).

When a body moves through a fluid medium with sufficient velocity to create a pocket of turbulence behind the body, an imbalance in the pressure surrounding the body—a pressure differential—is created (Figure 15-6). At the upstream end of the body where fluid particles meet the body head-on, a zone of relative high pressure is formed. At the downstream end of the body where turbulence is present, a zone of relative low pressure is created. Whenever a pressure differential exists, a force is directed from the region of high pressure to the region of low pressure. For example, a vacuum cleaner creates a suction force because a region of relative low pressure (the relative vacuum) exists inside the machine housing. This force, directed from front to rear of the body in relative motion through a fluid, constitutes form drag.

Several factors affect the magnitude of form drag, including the relative velocity of the body with respect to the fluid, the magnitude of the pressure gradient between the front and rear ends of the body, and the size of the surface area that is aligned perpendicular to the flow. Both the size of the pressure gradient and the amount of surface area perpendicular to the fluid flow can be reduced to minimize the effect of form drag on the human body. For example, streamlining the overall shape of the body reduces the magnitude of the pressure gradient. Streamlining minimizes the amount of turbulence created and hence minimizes the negative pressure that is created at the object’s rear (Figure 15-7). Assuming a more crouched body position also reduces the body’s projected surface area oriented perpendicular to the fluid flow.

Competitive cyclists, skaters, and skiers assume a streamlined body position with the smallest possible area of the body oriented perpendicular to the oncoming airstream. Even though the low-crouched aero-position assumed by competitive cyclists increases the cyclist’s metabolic cost as compared to an upright position, the aerodynamic benefit is an over tenfold reduction in drag (20). Similarly, race cars, yacht hulls, and some cycling helmets are designed with streamlined shapes. The aerodynamic frame and handlebar designs for racing cycles also reduce drag (8, 41).

Streamlining is also an effective way to reduce form drag in the water. The ability to streamline body position during freestyle swimming is a characteristic that distinguishes elite from subelite performers (9). Using a triathlon wet suit can reduce the drag on a competitor swimming at a typical triathlon race pace of 1.25 m/s by as much as 14%, because the buoyant effect of the wet suit results in reduced form drag on the swimmer (15, 48).

The nature of the boundary layer at the surface of a body moving through a fluid can also influence form drag by affecting the pressure gradient between the front and rear ends of the body. When the boundary layer is primarily laminar, the fluid separates from the boundary close to the front end of the body, creating a large turbulent pocket with a large negative pressure and thereby a large form drag (Figure 15-8). In contrast, when the boundary layer is turbulent, the point of flow separation is closer to the rear end of the body, the turbulent pocket created is smaller, and the resulting form drag is smaller.

The nature of the boundary layer depends on the roughness of the body’s surface and the body’s velocity relative to the flow. As the relative velocity of motion for an object such as a golf ball increases, changes in the acting drag occur (Figure 15-9). As relative velocity increases up to a certain critical point, the theoretical square law is in effect, with drag increasing with the square of velocity. After this critical velocity is reached, the boundary layer becomes more turbulent than laminar, and form drag diminishes because the pocket of reduced pressure on the trailing edge of the ball becomes smaller. As velocity increases further, the effects of skin friction and form drag grow, increasing the total drag. The dimples in a golf ball are carefully engineered to produce a turbulent boundary layer at the ball’s surface that reduces form drag on the ball over the range of velocities at which a golf ball travels.

Another way in which form drag can be manipulated is through drafting, the process of following closely behind another participant in speed-based sports such as cycling and automobile racing. Drafting provides the advantage of reducing form drag on the follower, since the leader partially shelters the follower’s leading edge from increased pressure against the fluid. Depending on the size of the pocket of reduced pressure behind the leader, a suctionlike force may also help to propel the follower forward. In swimming, the optimal drafting distance behind another swimmer in a swimming pool is 0–50 cm from the toes of the lead swimmer (10). Drafting has even been found to improve performance during a long-distance swim, particularly for faster and leaner swimmers (11).

Wave Drag

The third type of drag acts at the interface of two different fluids, for example, at the interface between water and air. Although bodies that are completely submerged in a fluid are not affected by wave drag, this form of drag can be a major contributor to the overall drag acting on a human swimmer, particularly when the swim is done in open water. When a swimmer moves a body segment along, near, or across the air and water interface, a wave is created in the more dense fluid (the water). The reaction force the water exerts on the swimmer constitutes wave drag.

The magnitude of wave drag increases with greater up-and-down motion of the body and increased swimming speed. The height of the bow wave generated in front of a swimmer increases proportionally with swimming velocity, although at a given velocity, skilled swimmers produce smaller waves than less-skilled swimmers, presumably due to better technique (less up-and-down motion) (47). At fast swimming speeds (over 3 m/s), wave drag is generally the largest component of the total drag acting on the swimmer (38). For this reason, competitive swimmers typically propel themselves underwater to eliminate wave drag for a small portion of the race in events in which the rules permit it. One underwater stroke is allowed following the dive or a turn in the breaststroke, and a distance of up to 15 m is allowed underwater after a turn in the backstroke. In most swimming pools, the lane lines are designed to minimize wave action by dissipating moving surface water.

While drag forces act in the direction of the free-stream fluid flow, another force, known as lift, is generated perpendicular to the fluid flow. Although the name lift suggests that this force is directed vertically upward, it may assume any direction, as determined by the direction of the fluid flow and the orientation of the body. The factors affecting the magnitude of lift are basically the same factors that affect the magnitude of drag:

In this equation, FL represents liftforce, CL is the coefficient of lift, ρ is the fluiddensity, Ap is the surface area against which lift is generated, and v is the relative velocity of a body with respect to a fluid. The factors affecting the magnitudes of the fluid forces discussed are summarized in Table 15-2.

Foil Shape

One way in which liftforce may be created is for the shape of the moving body to resemble that of a foil (Figure 15-10). When the fluid stream encounters a foil, the fluid separates, with some flowing over the curved surface and some flowing straight back along the flat surface on the opposite side. The fluid that flows over the curved surface is positively accelerated relative to the fluid flow, creating a region of relative high-velocity flow. The difference in the velocity of flow on the curved side of the foil as opposed to the flat side of the foil creates a pressure difference in the fluid, in accordance with a relationship derived by the Italian scientist Bernoulli. According to the Bernoulli principle, regions of relative high-velocity fluid flow are associated with regions of relative low pressure, and regions of relative low-velocity flow are associated with regions of relative high pressure. When these regions of relative low and high pressure are created on opposite sides of the foil, the result is a lift force directed perpendicular to the foil from the zone of high pressure toward the low-pressure zone.

Different factors affect the magnitude of the liftforce acting on a foil. The greater the velocity of the foil relative to the fluid, the greater the pressure differential and the lift force generated. Other contributing factors are the fluid density and the surface area of the flat side of the foil. As both of these variables increase, lift increases. An additional factor of influence is the coefficient of lift, which indicates a body’s ability to generate lift based on its shape.

The human hand resembles a foil shape when viewed from a lateral perspective. When a swimmer slices a hand through the water, it generates liftforce directed perpendicular to the palm. Synchronized swimmers use a sculling motion, rapidly slicing their hands back and forth, to maneuver their bodies through various positions in the water. The lift force generated by rapid sculling motions enables elite synchronized swimmers to support their bodies in an inverted position with both legs extended completely out of the water.

The semifoil shapes of projectiles such as the discus, javelin, football, boomerang, and frisbee generate some liftforce when oriented at appropriate angles with respect to the direction of the fluid flow. Spherical projectiles such as a shot or a ball, however, do not sufficiently resemble a foil and cannot generate lift by virtue of their shape.

The angle of orientation of the projectile with respect to the fluid flow—the angle of attack—is an important factor in launching a lift-producing projectile for maximum range (horizontal displacement). A positive angle of attack is necessary to generate a lift force (Figure 15-11). As the angle of attack increases, the amount of surface area exposed perpendicularly to the fluid flow also increases, thereby increasing the amount of form drag acting. With too steep an attack angle, the fluid cannot flow along the curved side of the foil to create lift. Airplanes that assume too steep an ascent can stall and lose altitude until pilots reduce the attack angle of the wings to enable lift (32).

To maximize the flight distance of a projectile such as the discus or javelin, it is advantageous to maximize lift and minimize drag. Form drag, however, is minimum at an angle of attack of 0˚, which is a poor angle for generating lift. The optimum angle of attack for maximizing range is the angle at which the lift/drag ratio is maximum. The largest lift/drag ratio for a discus traveling at a relative velocity of 24 m/s is generated at an angle of attack of 10˚ (19). For both the discus and the javelin, however, the single most important factor related to distance achieved is release speed (3, 21).

When the projectile is the human body during the performance of a jump, maximizing the effects of lift while minimizing the effects of drag is more complicated. In the ski jump, because of the relatively long period during which the body is airborne, the lift/drag ratio for the human body is particularly important. Research on ski jumping indicates that for optimal performance, ski jumpers should have a flattened body with a large frontal area (for generating lift) and a small body weight (for enabling greater acceleration) during takeoff. The effect of lift is immediate at takeoff, resulting in a higher initial vertical velocity than the jumper generates through impulse against the ramp surface (52). During the first part of the flight, jumpers should assume a small angle of attack to minimize drag (Figure 15-12). During the latter part of the flight, they should increase attack angle up to that of maximum lift. On smaller jumping hills where takeoff velocities are lower, jumpers should assume the attack angle for maximum lift earlier in the flight, because the effect of drag is not so great (40).

Other researchers investigating the ski jump have noted that the maximum length of the jump occurs neither at the body angle of attack for maximum lift nor at the attack angle for which the lift/drag is maximum, but somewhere between the two (Figure 15-13) (16). The ski jump consists of takeoff, flight, and landing, with body position during each phase influencing body position during the subsequent phase or phases. Research indicates that better performance is achieved when ski position is changed during the flight. Ski positions include the classic style, in which the skis are parallel; the V style, in which the skis are in a herringbone position in the frontal plane; and the flat V style, in which the skis are more flat in the sagittal plane than with the V style. Optimal performance results from initiating the jump in the classic style or the flat V style, and changing to the V style 1.3–1.6 s into the jump (23). This is true because the classic and flat V styles produce less drag than the V style in the first part of the jump, and the V style maximizes lift in the later part of the jump.

Magnus Effect

Spinning objects also generate lift. When an object in a fluid medium spins, the boundary layer of fluid molecules adjacent to the object spins with it. When this happens, the fluid molecules on one side of the spinning body collide head-on with the molecules in the fluid free-stream (Figure 15-14). This creates a region of relative low velocity and high pressure. On the opposite side of the spinning object, the boundary layer moves in the same direction as the fluid flow, thereby creating a zone of relative high velocity and low pressure. The pressure differential creates what is called the Magnus force, a lift force directed from the high-pressure region to the low-pressure region.

Magnus force affects the flight path of a spinning projectile as it travels through the air, causing the path to deviate progressively in the direction of the spin, a deviation known as the Magnus effect. When a tennis ball or table tennis ball is hit with topspin, the ball drops more rapidly than it would without spin, and the ball tends to rebound low and fast, often making it more difficult for the opponent to return the shot. The nap on a tennis ball traps a relatively large boundary layer of air with it as it spins, thereby accentuating the Magnus effect. The Magnus effect can also result from sidespin, as when a pitcher throws a curveball (Figure 15-15). The modern-day version of the curveball is a ball that is intentionally pitched with spin, causing it to follow a curved path in the direction of the spin throughout its flight path.

In 1982, a controversy between professional baseball players and scientists arose over the behavior of a pitched curveball. The players claimed that the path taken by a curveball was along a straight line until a certain critical point, at which the ball “broke” and suddenly curved. This appearance is enhanced when topspin is imparted to the ball, since the Magnus effect of topspin accentuates the effect of gravity. However, the actual path of a curveball is a smooth arc, which has been documented with high-speed movie film (2).

The extent to which a ball curves, or “breaks,” in the horizontal and vertical planes is dependent on the orientation of the spinning ball’s axis of rotation. If the axis of rotation is perfectly vertical, all of the Magnus effect occurs in the horizontal plane. Alternatively, if the axis of rotation is oriented horizontally, the Magnus effect is restricted to the vertical plane. Analysis of an actual baseball pitch thrown during the 1996 Olympic Games revealed an initial velocity of 35.9 m/s, an initial spin velocity of 18.5 rev/s, and the axis of rotation oriented nearly vertically at 85˚, resulting in a horizontal deviation of 39 cm (1). Curveballs thrown by major league pitchers spin as quickly as 27 revolutions per second and deviate horizontally as much as 40 cm over the pitcher-to-batter distance (49).

Soccer players also use the Magnus effect when it is advantageous for a kicked ball to follow a curved path, as may be the case when a player executing a free kick attempts to score. The “banana shot” consists of a kick executed so that the kicker places a lateral spin on the ball, curving it around the wall of defensive players in front of the goal (Figure 15-16).

The Magnus effect is maximal when the axis of spin is perpendicular to the direction of relative fluid velocity. Golf clubs are designed to impart some backspin to the struck ball, thereby creating an upwardly directed Magnus force that increases flight time and flight distance (Figure 15-17). When a golf ball is hit laterally off-center, a spin about a vertical axis is also produced, causing a laterally deviated Magnus force that causes the ball to deviate from a straight path. When backspin and sidespin have been imparted to the ball, the resultant effect of the Magnus force on the path of the ball depends on the orientation of the ball’s resultant axis of rotation to the airstream and on the velocity with which the ball was struck. When a golf ball is struck laterally off-center by a right-handed golfer, the ball unfortunately follows a curved path to one side—commonly known as a hook (to the left) or a slice (to the right).

Whereas a headwind slows a runner or cyclist by increasing the acting drag force, a tailwind can actually contribute to forward propulsion. Theoretical calculations indicate that a tailwind of 2 m/s improves running time during a 100 m sprint by approximately 0.18 s (54). A tailwind affects the relative velocity of a body with respect to the air, thereby modifying the resistive drag acting on the body. Thus, a tailwind of a velocity greater than the velocity of the moving body produces a drag force in the direction of motion (Figure 15-18). This force has been termed propulsive drag.

Analyzing the fluidforces acting on a swimmer is more complicated. Resistive drag acts on a swimmer, yet the propulsive forces exerted by the water in reaction to the swimmer’s movements are responsible for the swimmer’s forward motion through the water. The motions of the body segments during swimming produce a complex combination of drag and lift forces throughout each stroke cycle, and even among elite swimmers, a wide range of kinetic patterns during stroking have been observed (45). As a result, researchers have proposed several theories regarding the ways in which swimmers propel themselves through the water.

Propulsive Drag Theory

The oldest theory of swimming propulsion is the propulsive drag theory, which was proposed by Counsilman and Silvia (12) and is based on Newton’s third law of motion. According to this theory, as a swimmer’s hands and arms move backward through the water, the forwardly directed reaction force generated by the water produces propulsion. The theory also suggests that the horizontal components of the downward and backward motion of the foot and the upward and backward motion of the opposite foot generate a forwardly directed reaction force from the water.

When high-speed movie films of skilled swimmers revealed that swimmer’s hands and feet followed a zigzag rather than a straight-back path through the water, the theory was modified. It was suggested that this type of movement pattern enabled the body segments to push against still or slowly moving water instead of water already accelerated backward, thereby creating more propulsive drag. However, propulsive drag may not be the major contributor to propulsion in swimming.

Propulsive Lift Theory

The propulsive lift theory was proposed by Counsilman in 1971 (6). According to this theory, swimmers use the foillike shape of the hand by employing rapid lateral movements through the water to generate lift. The lift is resisted by downward movement of the hand and by stabilization of the shoulder joint, which translates the forward-directed force to the body, propelling it past the hand. The theory was modified by Firby (18) in 1975, with the suggestion that swimmers use their hands and feet as propellers, constantly changing the pitches of the body segments to use the most effective angle of attack.

A number of investigators have since studied the forces generated by the body segments during swimming. It has been shown that lift does contribute to propulsion and that a combination of lift and drag forces acts throughout a stroke cycle. The relative contributions of lift and drag vary with the stroke performed, the phase within the stroke, and the individual swimmer. For example, lift is the primary force acting during the breaststroke, whereas lift and drag contribute differently to various phases of the front crawl stroke (44). Drag generated by the swimmer’s hand is maximal when hand orientation is nearly perpendicular to the flow, and lift is maximal when the hand moves in the direction of either the thumb or the little finger (4).

Vortex Generation

Researchers have found a poor correlation between physiological and mechanical approaches to calculating propelling efficiency in swimming (5). This has led to the speculation that some unknown processes may play a role in swimming propulsion, with one possibility being the generation of vortices in the water by the swimmer. Vortex shedding has been found to play a role in the propulsion of both flying and swimming vertebrates and insects (17, 39). The generation of thrust in racing canoe and kayak paddling has also been described in terms of the mechanics of vortex–ring wakes (22). The observation that a swimmer performing the dolphin kick leaves behind a series of bound vortices, or columns of rotating water, has also been made (50). More research is needed to clarify the role of vortex generation in swimming propulsion.

Stroke Technique

Just as running speed is the product of stride length and stride rate, swimming speed is the product of stroke length (SL) and stroke rate (SR). Of the two, SL is more directly related to swimming speed among competitive freestyle swimmers (13). Comparison of male and female swimmers performing at the same competitive distances reveals nearly identical SRs, but longer SLs resulting in higher velocities for the males (37). At slower speeds, skilled freestyle swimmers are able to maintain constant, high levels of SL, with a progressive reduction in SL as exercise intensity increases due to local muscle fatigue (24). The same phenomenon has been observed over the course of distance events, with a general decrease in SL and swimming speed after the first 100 m (30). Research suggests that recreational freestyle swimmers seeking to improve swimming performance should concentrate on applying more force to the water during each stroke to increase SL, as opposed to stroking faster. Among backstrokers, although the ability to achieve a high swimming speed is related to SL at submaximal levels, increased speed is achieved through increased SR and decreased SL (25). Better performance in the breaststroke is associated with minimal vertical motion of the total body CG, but large range of hip vertical motion, during the stroke cycle (43).

Another technique variable of importance during freestyle swimming is body roll. In one study, competitive swimmers were found to roll an average of approximately 60˚ to the nonbreathing side (31). Research shows that body roll in swimming is caused by the turning effect of the fluidforces acting on the swimmer’s body. The contribution of body roll is important, since it enables the swimmer to employ the large, powerful muscles of the trunk rather than relying solely on the muscles of the shoulder and arm. It also facilitates the breathing action without any interruption of stroke mechanics (35). Body roll can influence the path of the hand through the water almost as much as the mediolateral motions of the hand relative to the trunk (31). In particular, an increase in body roll has been shown to increase the swimmer’s hand speed in the plane perpendicular to the swimming direction, thereby increasing the potential for the hand to develop propulsive lift forces (36). With increasing swimming speed, general body roll decreases, although trunk twist increases, allowing swimmers to benefit from the rolling of the upper trunk, while limiting the increase in the drag of the lower extremity (56).

The relative velocity of a body with respect to a fluid and the density, specific weight, and viscosity of the fluid affect the magnitudes of fluid forces. The fluid force that enables flotation is buoyancy. The buoyant force acts vertically upward, its point of application is the body’s center of volume, and its magnitude is equal to the product of the volume of the displaced fluid and the specific gravity of the fluid. A body floats in a static position only when the magnitude of the buoyant force and body weight are equal and when the center of volume and the center of gravity are vertically aligned.

Drag is a fluidforce that acts in the direction of the free-stream fluid flow. Skin friction is a component of drag that is derived from the sliding contacts between successive layers of fluid close to the surface of a moving body. Form drag, another component of the total drag, is caused by a pressure differential between the lead and trailing edges of a body moving with respect to a fluid. Wave drag is created by the formation of waves at the interface between two different fluids, such as water and air.

Lift is a force that can be generated perpendicular to the free-stream fluid flow by a foil-shaped object. Lift is created by a pressure differential in the fluid on opposite sides of a body that results from differences in the velocity of the fluid flow. The lift generated by spin is known as the Magnus force. Propulsion in swimming appears to result from a complex interplay of propulsive drag and lift.

For all problems, assume that the specific weight of fresh water equals 9810 N/m3 and the specific weight of seawater (salt water) equals 10,070 N/m3.

1. 1. A boy is swimming with an absolute speed of 1.5 m/s in a river where the speed of the current is 0.5 m/s. What is the velocity of the swimmer with respect to the current when the boy swims directly upstream? Directly downstream? (Answer: 2 m/s in the upstream direction; 1 m/s in the downstream direction)

2. 2. A cyclist is riding at a speed of 14 km/hr into a 16 km/hr headwind. What is the wind velocity relative to the cyclist? What is the cyclist’s velocity with respect to the wind? (Answer: 30 km/hr in the direction of the wind; 30 km/hr in the direction of the cyclist)

3. 3. A skier traveling at 5 m/s has a speed of 5.7 m/s relative to a headwind. What is the absolute wind speed? (Answer 0.7 m/s)

4. 4. A 700 N man has a body volume of 0.08 m3. If submerged in fresh water, will he float? Given his body volume, how much could he weigh and still float? (Answer: Yes; 784.8 N)

5. 5. A racing shell has a volume of 0.38 m3. When floating in fresh water, how many 700 N people can it support? (Answer: 5)

6. 6. How much body volume must a 60 kg person have to float in fresh water? (Answer: 0.06 m3)

7. 7. Explain the implications for flotation due to the difference between the specific weight of fresh water and the specific weight of seawater.

8. 8. What strategy can people use to improve their chances of floating in water? Explain your answer.

9. 9. What types of individuals may have a difficult time floating in water? Explain your answer.

10. 10. A beach ball weighing 1 N and with a volume of 0.03 m3 is held submerged in seawater. How much force must be exerted vertically downward to hold the ball completely submerged? To hold the ball one-half submerged? (Answer: 301.1 N; 150.05 N)

1. 1. A cyclist riding against a 12 km/hr headwind has a velocity of 28 km/hr with respect to the wind. What is the cyclist’s absolute velocity? (Answer: 16 km/hr)

2. 2. A swimmer crossing a river proceeds at an absolute speed of 2.5 m/s on a course oriented at a 45˚ angle to the 1 m/s current. Given that the absolute velocity of the swimmer is equal to the vector sum of the velocity of the current and the velocity of the swimmer with respect to the current, what is the magnitude and direction of the velocity of the swimmer with respect to the current? (Answer: 3.3 m/s at an angle of 32.6˚ to the current)

   

3. 3. What maximum averagedensity can a body possess if it is to float in fresh water? Seawater?

4. 4. A scuba diver carries camera equipment in a cylindrical container that is 45 cm long, 20 cm in diameter, and 22 N in weight. For optimal maneuverability of the container under water, how much should its contents weigh? (Answer 120.36 N)

5. 5. A 50 kg person with a body volume of 0.055 m3 floats in a motionless position. How much body volume is above the surface if the water is fresh? If the water is salt water? (Answer: 0.005 m3; 0.0063 m3)

6. 6. A 670 N swimmer oriented horizontally in fresh water has a body volume of 0.07 m3 and a center of volume located 3 cm superior to the center of gravity.

   • a. How much torque does the swimmer’s weight generate?

   • b. How much torque does the buoyant force acting on the swimmer generate?

   • c. What can the swimmer do to counteract the torque and maintain a horizontal position?

   • (Answer: 0; 20.6 N-m)

7. 7. Based on your knowledge of the action of fluidforces, speculate as to why a properly thrown boomerang returns to the thrower.

8. 8. Explain the aerodynamic benefits of drafting on a bicycle or in an automobile.

9. 9. What is the practical effect of streamlining? How does streamlining alter the fluidforces acting on a moving body?

10. 10. Explain why a curveball curves. Include a discussion of the aerodynamic role of the seams on the ball.

• Air and water are fluids that exert forces on the human body.

• The velocity of a body relative to a fluid influences the magnitude of the forces exerted by the fluid on the body.

• Atmospheric pressure and temperature influence a fluid’s density, specific weight, and viscosity.

• In order for a body to float, the buoyant force it generates must equal or exceed its weight.

• People who cannot float in swimming pools may float in Utah’s Great Salt Lake, in which the density of the water surpasses even that of seawater.

• Streamlining helps to minimize form drag.

• A ball projected with spin follows a trajectory that curves in the direction of the spin.

Kyle CR: Athletic clothing, Sci Am 254:104, 1986.

Presents an overview of the aerodynamic improvements that have been made in clothing for several different sports based on wind tunnel research on drag.

Morriss C and Bartlett R: Biomechanical factors critical for performance in the men’s javelin throw, Sports Med 21:438, 1996.

Reviews biomechanics research on the men’s javelin throw to promote understanding of how elite javelin throwers achieve success.

Toussaint HM, de Hollander AP, van den Berg C, and Vorontsov AR: Biomechanics of swimming. In Garrett WE and Kirkendall DT: Exercise and sport science, Philadelphia, 2000, Lippincott Williams & Wilkins.

Provides a comprehensive review of literature on the effects of drag in swimming and on swimming propulsion techniques and theories.

Townend MS: Mathematics in sport, New York, 1984, John Wiley & Sons.

Includes a chapter on sailing that provides a technical analysis of the aerodynamics and hydrodynamics of sailing and windsurfing.

Circulation and the Magnus Effect

• www.phys.virginia.edu/classes/311/notes/aero/node2.html
• Shows a diagram and includes discussion of the Magnus force with practical examples.

Drag Force on a Sphere

• www.ma.iup.edu/MathDept/Projects/CalcDEMma/drag/drag0.html
• Provides links to pages on the graph of the drag coefficient versus Reynolds number and two models for drag force.

Magnus Force

• www.rawbw.com/~xmwang/physDemo.html#magnus
• Provides links to a number of graphical animations of the effect of the Magnus force.

NASA: Lift from Pressure

• www.grc.nasa.gov/WWW/K-12/airplane/right1.html
• Provides narrative, definitions of related terms, and slides illustrating lift concepts.

NASA: Lift to Drag Ratio

• www.grc.nasa.gov/WWW/K-12/airplane/ldrat.html
• Provides narrative, definitions of related terms, and slides illustrating lift/drag ratio concepts.

NASA: Relative Velocities

• www.grc.nasa.gov/WWW/K-12/airplane/move2.html
• Provides narrative, definitions of related terms, and slides illustrating relative velocity concepts.

NASA: What Is Drag?

• www.grc.nasa.gov/WWW/K-12/airplane/drag1.html
• Provides narrative, definitions of related terms, and slides illustrating drag concepts.

Open Door Website: Relativity

• www.saburchill.com/physics/chapters/0083.html
• Provides description of a quantitativerelative velocity problem with entertaining graphics.

Physics Classroom: Relative Velocity

• www.physicsclassroom.com/Class/vectors/U3L1f.html
• Includes explanation, graphics, and animations demonstrating relative velocity.

Relative Velocity Applet

• www.math.gatech.edu/~carlen/2507/notes/classFiles/partOne/RelVel.html
• Includes an interactive application that enables control of two moving points and provides the ability to graph the absolute and relative motions of the points.

Tennis: The Magnus Effect

• http://wings.avkids.com/Tennis/Book/magnus-01.html
• Provides a description and animated drawing of the Magnus effect on a spinning tennis ball.

U.S. Centennial of Flight Commission

• www.centennialofflight.gov/essay/Dictionary/four_forces/DI24.htm
• Provides an illustrated discussion about the forces acting on an airplane.