Chapter 11. The Description of Human Motion

At the conclusion of this chapter, the student should be able to:

• 1. Name the kinds of motion experienced by the human body, and describe the factors that cause and modify motion.
• 2. Name and properly use the terms that describe linear and rotary motion: position, displacement, distance, speed, velocity, and acceleration.
• 3. Explain the interrelationships that exist among displacement, velocity, and acceleration, and use the knowledge of these interrelationships to describe and analyze human motion.
• 4. Describe the behavior of projectiles, and explain how angle, speed, and height of projection affect that behavior.
• 5. Describe the relationship between linear and rotary movement, and explain the significance of this relationship to human motion.
• 6. Identify the critical kinematic components that would be used to fully describe the skillful performance of a selected motor task.

If we are to understand the movements of the human musculoskeletal system and the objects put into motion by this system, we need first to turn our thoughts to the concepts of motion itself. What is motion? What determines the kind of motion that will result when an object or a part of the human body is made to move? How is motion described in mechanical terms? How do these generalities about motion apply to movements of the musculoskeletal system? Indeed, how does one know that motion is occurring?

Relative Motion

Motion is the act or process of changing place or position with respect to some reference object. Whether a body is at rest or in motion depends totally on the reference, global or local. When a person is walking down the street or riding a bicycle or serving a tennis ball, it seems obvious that movement is involved. Less obvious is the motion status of the sleeping passenger in a smoothly flying plane or of an automobile parked at a curb. If the earth is the reference point, all but the parked car are in motion relative to the earth, and even the parked car is in motion if the reference point is the sun. On the other hand, if the bicycle is the reference point, the person riding it is at rest relative to the bicycle, and the sleeping passenger is at rest with respect to anything in the plane. The relative motion of each is defined in relation to the specific reference object or point. It is possible, therefore, to be at rest and in motion at the same time relative to different reference points. The sleeping passenger is at rest relative to the plane and in motion relative to the earth. The relative motion of two bodies depends entirely on their relative velocities through space. Two joggers running at 8 km/hr in the same direction are at rest with respect to each other. However, if one jogs at 8 km/hr and the other at 10 km/hr, the slower jogger would be considered to be at rest with respect to the faster but the faster would be in motion both with respect to the slower runner and to the earth.

Cause of Motion

It is difficult to think of motion without visualizing a specific object in the act of moving. If we did not actually see how it changed from a stationary condition to a moving one, we might wonder what caused it to be set in motion. Did someone pull on it, or push against it, or perhaps blow on it or even attract it with a magnet? What are these assumed causes of motion? Without exception, each cause of motion is a form of force. Force is the instigator of movement. If we see an object in motion, we know that it is moving because a force has acted on it. We know, too, that the force must have been sufficiently great to overcome the object’s inertia, or resistance to motion, for unless a force is greater than the resistance offered by the object, it cannot produce motion. We can push against a stone wall all day without moving it so much as 1 millimeter, but a bulldozer can knock down the wall at the first impact. The magnitude of the force relative to the magnitude of the resistance is the determining factor in causing an object to move.

Kinds of Motion

What are the ways in which an object may move? The hand moves in an arc when the forearm turns at the elbow joint and the neighboring joints are held motionless. A hockey puck may slide across the ice without turning. On the other hand, it may revolve as it slides. A figure skater spins in place. Arrows, balls, and jumpers move through the air in an arc known as a parabola. As we note the different ways in which objects move, we are impressed with the almost limitless variety in the patterns of movement. Objects move in straight paths and in curved paths; they roll, slide, and fall; they bounce; they swing back and forth like a pendulum; they rotate about a center, either partially or completely; and they frequently rotate at the same time that they move as a whole from one place to another. Although the variety of ways in which objects move appears to be almost limitless, careful consideration of these ways reveals that there are, in actuality, only two major classifications of movement patterns. These are translatory, or linear, and rotary, or angular. Either an object moves in its entirety from one place to another, or it turns about a center of motion. Sometimes it does both simultaneously.

Translatory (Linear) Movement

This kind of movement is termed translatory because the object is translated as a whole from one location to another. Translatory movement is commonly called linear motion and is further classified as rectilinear or curvilinear. Rectilinear motion is the straight-line progression of an object as a whole with all its parts moving the same distance in the same direction at a uniform rate of speed. The child on the sled in Figure 11.1, a water skier pulled by a boat, or a bowling ball moving in a straight path are examples of rectilinear motion.

Curvilinear motion refers to all curved translatory movement; that is, the object moves in a curved pathway. The paths of a ball or any other projectile in flight, the wrist during the force phase in bowling (Figure 11.2), or a skier in a sweeping turn are all examples of curvilinear motion.

A special form of curvilinear motion, which on the surface does not appear to be translatory, is that called circular motion. This type of motion occurs when an object moves along the circumference of a circle, that is, a curved path of constant radius. The logic for calling this type of motion linear relates to the fact that it occurs when an unbalanced force acts on a moving body to keep it in a circle. If that unbalanced force stops acting on the object and the object is free to move, it will move in a linear path tangent to the direction in which it is moving at the moment of release. A classic example is an object tied to the end of a string and swung in circles around the head. When the object is released, it will fly off in a straight line. The hammer in the hammer throw follows a circular path until it is released, at which time it flies along a curvilinear path until it lands. The path of a ball held in the hand as the arm moves around in windmill fashion is another example of circular motion. If the ball is released during the motion, it will fly off at a tangent and continue in a straight line until gravity forces it into a curved path. Other examples of bodies in circular motion are the gondola on a moving Ferris wheel or the knot on the ring of a spinning lariat.

Rotary (Angular) Motion

This kind of motion is typical of levers and of wheels and axles. Rotary, or angular, motion occurs when any object acting as a radius moves about a fixed point. The distance traveled may be a small arc or a complete circle and is measured as an angle, in degrees. Most human body segment motions are angular movements in which the body part moves in an arc about a fixed point. The axial joints of the skeleton act as fixed points for rotary motion in the segments. The arm engages in rotary motion when it moves in windmill fashion about a fixed point or axis in the shoulder. The head’s motion in the act of indicating “no,” the lower leg in kicking a ball, or the hand and forearm in turning a doorknob are all examples of rotary, or angular, motion. In each instance the moving body segment may be likened to the radius of a circle. The arm moving in windmill fashion and the lower leg and foot in kicking are the radii (Figure 11.3). In the “no” action of the head and in the doorknob being turned by the forearm and hand, the radius is perpendicular to the long axis running vertically through the middle of the head and lengthwise through the middle of the forearm and hand, respectively. These movements are not to be confused with circular motion. Circular motion describes the motion of any point on the radius, whereas angular motion is descriptive of the motion of the entire radius. When a ball is held in the hand as the arm moves in a windmill fashion, the ball is moving with circular motion, while the arm acts as a radius moving with angular motion about the fixed point of the shoulder.

Other Movement Patterns

Reciprocating motion denotes repetitive movement. The use of the term is ordinarily limited to repetitive translatory movements, as illustrated by a bouncing ball or the repeated blows of a hammer, but technically it includes all kinds. The term oscillation refers specifically to repetitive movements in an arc. Familiar examples of this type of movement are seen in the pendulum, metronome, and playground swing.

Often an object displays a combination of rotary and translatory movement. This is sometimes referred to as general motion. The bicycle, automobile, and train move linearly as the result of the rotary movements of their wheels. Likewise, people, as they walk or run down the street, experience translatory motion because of the angular movement of their body segments. The angular motions of several segments of the body are frequently coordinated in such a way that a single related segment will move linearly. This is true in throwing darts, in shot putting, and in a lunge in fencing (Figure 11.4). Because of the angular motions of the forearm and upper arm, the hand travels linearly and thus is able to impart linear force to the dart, to the shot prior to the release, and to the foil.

Kinds of Motion Experienced by the Body

The human body experiences all kinds of motion. Because most of the joints are axial, the body segments must undergo primarily angular motion (Figure 11.5). A slight amount of translatory motion is seen in the gliding movements of the plane or irregular joints, but these movements are negligible in themselves. They occur chiefly in the carpal and tarsal joint and in the joints of the vertebral arches in conjunction with angular movements in neighboring axial joints.

The body as a whole experiences rectilinear movement when it is acted on by the force of gravity, as in coasting down a hill (Figure 11.1) or in a free fall (Figure 11.6), and likewise when acted on by an external force, as in water skiing (Figure 11.7). It experiences general motion in forward and backward rolls on the ground and in somersaults in the air, and rotary motion in twirling on ice skates. It experiences curvilinear translatory motion in diving and jumping, and it experiences reciprocating motion when swinging back and forth on a swing or a bar.

Factors that Determine the Kind of Motion

Thus far we have considered the cause of motion and the various kinds of motion on the basis of movement patterns or paths. Now we must turn to another question. What determines the kind of motion that will result when an object is made to move? The best way for the student to discover the answer is to produce each kind of motion and then analyze what was done to obtain the desired motion.

To make an object move linearly, we discover that it must be free to move and that either we must apply force uniformly against one entire side of the object or we must apply it directly in line with the object’s center of gravity. The object will move in a straight line provided it does not meet an obstacle or resistance of some sort. If its edge hits against another object or encounters a rough spot, the moving object will turn about its point of contact with the interfering obstacle. If we attempt to push a tall cabinet across a supporting surface that provides excessive friction, such as a cement floor, the cabinet will tip, even though we place our hands exactly in line with the cabinet’s center of gravity and push in a horizontal direction. To move it linearly, it is necessary to apply the push lower than the cabinet’s center of gravity to compensate for the friction.

If one part of an object is “fixed,” rotary motion will occur when sufficient force is applied on any portion of the object that is free to move. A lever undergoes rotary motion because, by definition, some portion of it remains in place. To move an object in the manner of a lever, it is necessary to provide a “fulcrum” or an axis and to apply force to the object at some point other than at the fulcrum. Thus, if rotary motion of a freely movable object is desired, it is necessary to apply force to it “off center” or to provide an “off center” resistance that will interfere with the motion of part of the object.

Reciprocating motion is caused by a uniform repetition of opposing force applications, and the oscillation of a pendulum is produced by repeated applications of gravitational force to a suspended object that is free to move back and forth and that is in any position other than its resting position.

In summary, the kind of motion that will be displayed by a moving object depends primarily on the kind of motion permitted in that particular kind of object. If it is a lever, it is permitted only angular motion; if it is a pendulum, oscillatory motion; and so on. If it is a freely movable object, it is permitted either translatory or rotary motion, depending on the circumstances. These circumstances include the point at which force is applied with reference to the object’s center of gravity, the environmental pathways of movement available to the object, and the presence or absence of additional external factors that modify the motion.

Factors Modifying Motion

Motion is usually modified by a number of external factors, such as friction, air resistance, and water resistance. Whether these factors are a help or a hindrance depends on the circumstances and the nature of the motion. The same factor may facilitate one form of motion, yet hinder another. For instance, friction is a great help to the runner because maximum effort may be exerted without danger of slipping; on the other hand, friction hinders the rolling of a ball, as in field hockey, golf, and croquet. Again, wind or air resistance is indispensable to the sailboat’s motion, but unless it is a tailwind, it impedes the runner. Likewise, water resistance is essential for propulsion of the body by means of swimming strokes and of boats through the use of oars and paddles, yet at the same time it hinders the progress of both the swimmer and the boat, especially if these pre-sent a broad surface to the water. For this reason, swimmers keep the body level, and designers plan streamlined boats. One of the major problems in movement is to learn how to take advantage of these modifying factors when they contribute to the movement in question and how to minimize them when they are detrimental to the movement. A more detailed discussion of forces influencing motion is presented in Chapters 12 and 13.

The motion of the segments of the body is also modified by anatomical factors. These include friction in the joints (minimized by synovial fluid), tension of antagonistic muscles, tension of ligaments and fasciae, anomalies of bone and joint structure, atmospheric pressure within the joint capsule, and the presence of interfering soft tissues. Except for the limitations because of fat or muscle bulk, these modifying factors are classified as internal resistance.

Motion has been defined as the act or process of changing place or position with respect to some reference point. Thus, to talk about motion, a starting point must be identified. Once this is done, the resultant motion, regardless of whether it is translatory or rotary, may be characterized according to the distance and direction away from the starting point, the speed of the movement, and any change in speed that may occur. This kind of motion study is called kinematics. Motion is described in terms of displacement, velocity, and acceleration with no consideration of or reference to the forces that cause or modify the motion. Linear kinematics is concerned with translatory motion and angular kinematics with rotary motion.

Linear Kinematics

Distance and Displacement

The distance an object is removed from a reference point is called its displacement. Displacement does not indicate how far the object travels in going from point A to point C. It only indicates the final change of position. A person who walks north for 3 kilometers to point B and then east for 4 kilometers to point C has walked a distance of 7 kilometers, but the displacement with respect to the starting point is only 5 kilometers (Figure 11.8). Similarly, a basketball player who runs up and down the court several times has traveled a considerable distance, but the displacement with respect to one of the end lines may be zero. Or consider the poor golfer who, blinded by the late afternoon sun, hits the ball so erratically and frequently that the route to the green, 450 yards away, crosses and recrosses the fairway many times. Regardless of the zigzag path to the green and the many changes of direction needed to get there, the ball’s displacement is the straight-line distance from the tee to the green.

Displacement is a vector quantity having both magnitude and direction. It is not enough to indicate only the amount of positional change. That alone would be distance, a scalar quantity. The direction of the vector must also be defined. When the golfer finally reaches the green, the displacement from the hole to the green is 450 yards west. And the walker’s displacement in Figure 11.8 is 5 km, in a northeast direction.

Speed and Velocity

Speed and velocity are frequently used to describe how fast an object is moving. These terms are often used interchangeably, but in fact there is a significant difference. Speed is related to distance and velocity to displacement. Speed tells how fast an object is moving—that is, the distance an object will travel in a given time—but it tells nothing about the direction of movement.

Examples of speed measurements are a car traveling at 7 km/hr, the wind blowing at 60 mph, a ball thrown with a speed of 30 m/sec, or a sprinter running at 10 m/sec.

Velocity, on the other hand, involves direction as well as speed. Speed is a scalar quantity, whereas velocity is a vector quantity. In many activities this difference is of no concern, but in others it is of extreme importance. The speed of a football player carrying the ball may be impressive, but if not directed toward the opponent’s goal, it is not providing yardage for a first down. Although the speed may be great, the velocity in the desired direction may indeed be zero. Velocity is speed in a given direction. It is the amount of displacement per given unit of time. This is the same as saying that velocity is the rate of displacement, or

In the diagrams in Figure 11.9, displacement values (s) are represented on the y axis and the time values (t) are on the x axis. If displacement values are plotted to correspond with their time values, the line formed by connecting these plotted values represents the rate of displacement or velocity (v). When the rate of displacement does not change—that is, when the distance and direction traveled is the same for each equal time period—the velocity is constant, and the velocity line on the diagram is a straight line. In Figures 11.9a, b, and c the velocity is constant, but in Figure 11.9d the curved line indicates that the rate of displacement changes, and therefore the velocity is not constant. When there is greater displacement per unit of time, the velocity increases, as does the slope of the velocity line in the diagram. Figure 11.9b shows the fastest velocity and c the slowest. In d, the displacement starts at a slow rate and then increases. If a, b, c, and d represent runners on a straight track, a, b, and c would each be running at a constant but different velocity, with b’s velocity the fastest and c’s the slowest. Runner d starts out at a slow velocity but increases the rate of displacement until the resultant velocity is the fastest of all four.

Where velocity is constant, as in a, b, and c, the motion is said to be uniform. When the amount of displacement per unit of time varies, nonuniform motion occurs. Uniform motion is not a common characteristic of human motion because most human movements are likely to have many variations in the rate of displacement. When the velocity of human motion is given, it is usually an average velocity that tells only the total displacement occurring in a stated period of time. Although a long-distance runner who ran the Boston marathon, a distance of 26 miles, 385 yards, in 2.5 hours had an average velocity of 10.4 mph, it is doubtful that the velocity was uniformly 10.4 mph throughout the run. If one were to record the time at which the runner passed frequent and equally spaced distance points along the route, a displacement–time graph could be prepared to show the variations in the runner’s speed at various points in the course of the race. This kind of information can be quite useful in helping a coach or participant analyze the performance and strategy of the race and plan changes where needed. The narrower the distance intervals used, the greater is the possibility that critical variations in speed will become apparent. The use of motion analysis systems permits a similar analysis of brief and fast events. The distance and time data necessary for graphing and analyzing the motion patterns are obtained indirectly from the video or digital record.

In equation form, average velocity is

The symbol s represents displacement, and t represents time. The average velocity of a tennis ball served 19 meters in 0.35 second is 19 divided by 0.35, or 54.3 m/sec in the direction of the service court.

Acceleration

When velocity changes, its rate of change is called acceleration. A sprint runner has an initial velocity of 0 m/sec. When the gun signals the beginning of a race, the sprinter’s velocity begins to change by increasing. The rate of change in velocity is acceleration. Acceleration may be positive or negative. An increase is considered positive, and a decrease such as slowing down at the end of the race is negative. Negative acceleration is also called deceleration.

In equation form, acceleration is expressed as

where ā represents average acceleration, vf is the final velocity and vi is the initial velocity. In other words, acceleration is any change in velocity divided by the time interval over which that change occurred.

In the example of the sprint runner, a graph of the sprinter’s velocity throughout the race can be used to illustrate acceleration (Figure 11.10). As the sprinter is waiting in the starting blocks, the velocity is zero. In section a of the race, the sprinter changes from a velocity of 0 to a velocity of 9 m/sec after the first 5.6 seconds. Because this is an increase in velocity, this is positive acceleration. The acceleration for this phase of the race, the rate of change in velocity, equals the difference between the final velocity (9 m/sec) and the initial velocity (0 m/sec) divided by the time interval (5.6 sec), or 1.8 m/sec2. In section b of the race the velocity does not change but remains at a constant 9 m/sec. Because there is no change in velocity, the acceleration for this phase would be zero. To prove this, we use the equation

In section c the sprinter increases velocity again just before the finish. In this phase the acceleration is not constant, so calculating an average velocity would not produce a true representation of the changes in velocity that are occurring. It is more accurate to calculate several instantaneous accelerations to plot a curve. By visually examining the curve, it can be seen that the acceleration at the beginning of this short phase is greater than the acceleration at the end of the phase. Section d of the race starts as the sprinter crosses the finish line. At this point the velocity is 10 m/sec. The sprinter now decreases velocity to come to a stop (0 m/sec) after the race. This slowing down represents negative acceleration, or deceleration.

Because velocity is always displacement divided by time and acceleration is velocity divided by time, acceleration is really displacement divided by time divided by time, and the units for measurement must reflect this. The time unit must appear twice in acceleration units. The logic for this is apparent when the units of m/sec are used for velocity in the equation for average acceleration:

After the subtraction is completed this equation becomes

or, as commonly written, m/sec/sec or m/sec2. Thus, the average acceleration of the runner in the example is 1.6 m/sec2.

One usually thinks of acceleration in terms of a change in the amount of distance covered in equal units of time. Acceleration also occurs when, although the speed remains constant, there is a change in direction. The example given for Figure 11.10b is that of a runner keeping a steady pace on a straight track with no acceleration. If the runner, still running at the same speed, shifts to a circular track, acceleration occurs because of the change in direction, and the velocity–time graph would look more like Figure 11.10c.

Uniformly Accelerated Motion

When the acceleration rate is constant, the velocity change is the same during equal time periods. Under these conditions motion is said to be uniformly accelerated. This type of acceleration does not occur with great frequency because the change in velocity of bodies in motion is usually irregular and complicated. However, one common type of uniform acceleration is important in sport and physical education, the acceleration of freely falling bodies.

Neglecting air resistance, objects allowed to fall freely will speed up or accelerate at a uniform rate owing to the acceleration of gravity. Conversely, objects projected upward will be slowed at a uniform rate that is also due to the acceleration of gravity. The value for the acceleration of gravity changes with different locations on the earth’s surface, but for most of the United States this value can be considered to be 32 ft/sec2 or 9.80 m/sec2. Regardless of its size or density, a falling object will be acted on by gravity so that its velocity will increase 9.8 m/sec each second it is in the air. A dropped ball starting out with a velocity of 0 m/sec will have a velocity of 9.8 m/sec at the end of 1 second, 19.6 m/sec at the end of 2 seconds, 29.4 m/sec at the end of 3 seconds, and so on. A second ball weighing twice as much will fall with exactly the same acceleration. It too will have a velocity of 29.4 m/sec at the end of 3 seconds. Of course, this example does not consider the resistance or friction of air, which can be appreciable. The lighter the object, the more it is affected. After an initial acceleration, light objects such as feathers or snowflakes may stop accelerating entirely and fall at a constant rate. Consider, for instance, the difference between the behavior of a badminton shuttle and a golf ball when dropped from a height.

The denser and heavier the free-falling object, the less air friction affects it, especially if the distance of the fall is not too great. Even heavy objects, such as sky divers falling from great distances, eventually reach a downward speed large enough to create an opposing air resistance equal to the accelerating force of gravity. When this happens, the diver no longer speeds up but continues to fall at a steady speed. This speed is called terminal velocity and amounts to approximately 120 mph (53 m/sec) for a falling sky diver. With the parachute open, the diver’s velocity decreases to 12 mph steady velocity.

Laws of Uniformly Accelerated Motion

In spite of the reality of air resistance, much can be learned about the nature of free-falling bodies and uniform acceleration through a knowledge of the laws of uniformly accelerated motion. Because the acceleration of gravity is constant, the distance traveled by a freely falling body, as well as its downward velocity, can be determined for any point in time by application of these laws. Expressed in equation form, they are

Galileo’s experiments with inclined planes enabled him to work out these equations. They apply to any type of linear motion in which acceleration is uniform. Their specific application to the effect of gravity on freely falling objects is presented in Table 11.1. If the initial velocity (vi) is zero, as it is when an object is allowed to fall freely from a stationary position, the equations may be simplified:

The student may also discover that some authors, when applying these equations specifically to gravity, replace a, the symbol for acceleration, with g, the symbol for gravity.

The time an object takes to rise to the highest point of its trajectory is equal to the time an object takes to fall to its starting point. Similarly, the release speed and landing speed are the same. Other than the fact that the directions are reversed, the upward flight is a mirror image of the downward flight. Proof that the release velocity and landing velocity are equal in amount but opposite in direction can be shown mathematically by substitution of values in the motion equations. Following vector conventions, velocities upward are positive and those downward are negative. Thus the acceleration of gravity is treated as a negative value.

Example Assuming that a ball is thrown upward so that it reaches a height of 5 meters before starting to fall, what is its initial velocity as it leaves the hand? What is its final velocity as it lands in the hand?

Upward Thrown Velocity

Downward Landing Velocity

Projectiles

An object that has been given an initial velocity and then allowed to move in free fall under the influence of gravity is a projectile. Balls that are thrown, kicked, or hit; javelins; bullets or missiles; and jumpers, divers, and gymnasts while in the air are all examples of projectiles.

An object or a body is projected into the air for any of several reasons. In the case of the diver or the gymnast, the purpose of the projection is to gain maximum time in the air, or time of flight. The longer the time of flight the athlete can produce, the greater the number of acrobatic moves that can be performed. In other activities, a decreased time of flight may serve to deceive or avoid an opponent, as in the volleyball spike, or a smash in tennis, or an onside kick in football. Projectiles may also be released for the purpose of producing maximum horizontal displacement. The long jumper, the discus thrower, the shotputter, and the batter in baseball are all examples of projection of an object for distance. Maximum displacement may also be in the vertical direction. Projections for maximum vertical displacement include such activities as the high jump and the pole vault. Projection of a body or an object for maximum accuracy is the purpose of actions such as shooting in basketball or soccer, passing, archery, or golf. When accuracy is the primary concern, a compromise often must be made between horizontal displacement and time in the air.

Once released, projectiles follow a predictable path. If air resistance is ignored because it is considered negligible, this path will be a parabola. The characteristic parabolic path of a projectile is the result of the constant downward force of gravity. This means that gravity will decelerate any upward motion at a rate of 9.8 m/sec2 or will accelerate any downward motion at the same rate (Figure 11.11). All objects in free fall have the same downward acceleration whether they start from a resting position with a drop or fall or have been given some initial velocity.

Two forces then, are acting on a projectile: the projecting force and gravity. The projecting force is a vector quantity that may act at any desired angle, depending on the purpose of the projection. The application of the projecting force produces an initial velocity in the object at some angle of projection. Because this initial release velocity is a vector quantity, it can be resolved into two component velocities, one vertical and one horizontal. The vertical component of velocity, being a vector parallel to gravity, will be directly affected by gravity. The horizontal component will not. Gravity is also a vector force that always acts in a vertical, downward direction. This downward force of gravity acts completely independently of any horizontal component of the projecting force (Figure 11.12).

If an object or a body is projected with only a horizontal velocity, then gravity, as the second force, will still act to cause that object to fall. If one object falls freely from rest at the same time that another is projected horizontally from the same height, both objects will hit the ground at the same time. However, they will hit in different places. The dropped object will land immediately below the point of release, whereas the projected object will land some distance away. Gravity has acted on both objects equally, giving them equal vertical velocities in the downward direction, so they will fall at the same rate and land at the same time. The difference in landing points is the result of the horizontal velocity possessed by the projected object. In the time it took both objects to fall, the horizontal velocity of the projected object carried that object some distance from the point of release. This distance can be calculated using the velocity equation

If, for instance, these two objects were balls released at a height of 2 meters, they would take 0.64 second to fall to the ground. If one was projected horizontally with a velocity of 20 m/sec, it would strike the ground at a distance of 12.8 meters from the release point (s = 20 m/sec * 0.64 sec) (Figure 11.13).

To change the time an object is in the air, the velocity produced by projection must have some vertical component. This may be an upward component, opposing gravity, or a downward component being added to gravity. The time an object is in the air may also be varied by altering the height of release. If the balls in the previous examples were released from a height of 3 meters, both balls would be in the air for .78 seconds. Using the velocity equation again, it can be determined that the horizontally projected ball would now travel 15.6 meters from the point of release (s = 20 m/sec * .78 sec = 15.6 meters).

An object that is projected with only upward velocity will be decelerated by gravity until it reaches a velocity of zero. At this point, it will start to drop back toward the release point, accelerating as it falls. When the object reaches the release point, it will possess the same velocity it was given at release. The time required to reach the highest point will be equal to the time it takes to fall back to the height from which it was released.

More often than not, objects put into flight will be projected in some direction other than exactly vertical or horizontal. A projectile of this type has both horizontal and vertical components of the initial velocityvector. Again, these two component velocities are considered independently. The horizontal component of velocity remains constant following release (if air resistance is neglected), as no force is available to change this velocity. The vertical component of velocity will be subject to the uniform acceleration of gravity. When the object is projected with some upward angle, gravity will act to decelerate the object to zero vertical velocity and then accelerate the object again as it falls downward. During this period of vertical deceleration and acceleration, the object is also undergoing constant horizontal motion. This combination of these two independent factors produces the parabolic flight path of the projectile as portrayed in Figure 11.14.

The horizontal distance an object will travel in space depends on both its horizontal velocity and the length of time the object is in the air, or time of flight. The time of flight depends on the maximum height reached by the object, and that, in turn, is governed by the vertical velocity imparted to the object at release. Thus the horizontal distance an object will travel depends on both the horizontal and vertical components of velocity. As will be remembered from the earlier discussion of vectors, the magnitudes of these two components will be determined by the magnitude of the initial projection velocity vector and by the angle that indicates the direction of this vector, referred to as the angle of projection. With this in mind, it can be seen that a projectile with a low angle of projection will have a relatively high horizontal velocity in relation to the vertical velocity. The low vertical velocity does little to resist the pull of gravity, which therefore requires very little time to decelerate the object to a vertical velocity of zero and start the drop back down. In this instance, vertical distance is low, and therefore time of flight is short, allowing little time for horizontal travel. On the other hand, if the angle of projection is large, it takes longer for the object to decelerate to zero velocity, allowing a much longer time of flight. In this instance, however, there is little horizontal velocity, so little distance can be covered in the time available. Thus it would seem that the optimum angle of projection would be a 45-degree angle, with equal magnitudes for the horizontal and vertical components. In fact, the actual optimal angle of projection depends on several factors, including purpose of the projection. A 45-degree angle of projection will maximize horizontal distance only if release height and landing height are the same. In this case, the object will approach the landing at approximately the same angle as that at which it was projected (Figure 11.15).

If an object is projected from above the ground, as in many throwing events, a lower angle of projection may produce optimum results. This is because the object thrown will have a somewhat increased time of flight as it covers the extra distance between the height of release and the ground as it falls. With this increased time of flight, a slightly reduced vertical velocity and a slightly increased horizontal velocity will usually be optimal. The greater the difference between release height and landing height, the lower the angle of projection needs to be. If initial velocity can be increased, the optimum angle can also be increased back toward a 45-degree angle. From this discussion, then, it can be concluded that speed of release, angle of projection, and height of release are the three factors that control the range of a projectile.

Angular Kinematics

Angular kinematics is very similar to linear kinematics because it is also concerned with displacement, velocity, and acceleration. The important difference is that the displacement, velocity, and acceleration are related to rotary rather than to linear motion, and although the equations used to show the relationships among these quantities are quite similar to those used in linear motion, the units used to describe them are different (Table 11.2).

Angular Displacement

The human skeleton is made up of a system of levers that by definition are rigid bars that rotate about fixed points when force is applied. When any object acting as a rigid bar moves in an arc about an axis, the movement is called rotary, or angular, motion. An attempt to describe angular motion in linear units presents real problems. As an object moves in an arc, the linear displacement of particles spaced along that lever varies. Particles near the axis have a displacement in inches, meters, feet, or centimeters that is less than those farther away. For example, in the underarm throw pattern, the hand moves through a greater distance than the wrist and the wrist a greater distance than the elbow. Rotary motion needs rotary units to describe it. As might be expected, these units relate to the units of a circle and the fact that the circumference of a circle C is equal to 2πr, where r is the radius and π is a constant value of 3.1416.

There are three interchangeable rotary, or angular, units of displacement: degrees, revolutions, and radians. One radian is the same as 57.3 degrees, and one full revolution is the same as 360 degrees or 2π radians of displacement. The word revolution is not stated, but it is understood when we say that a diver executed a 1 1/2 somersault tuck. The dive could also be described as a tuck somersault of 540 (360 + 180) degrees or 3π radians. Degrees are used most frequently in the measurement of angles, but radians, the term favored by engineers and physicists, are the units most often required in equations of angular motion. The advantage of using radians is that they have no units and, therefore, may be used in equations with linear kinematic terms, as we will see. A radian is the angle at which the subtended arc of a circle is equal to the radius. The result of having 2π radians in a circle is that there are 6.28 radians in 360 degrees. If the circle is divided by the 6.28 radians it contains, it can be seen that each radian is 57.3 degrees. The symbol for angular displacement is the Greek letter θ (theta).*

*For a review of the geometry of circles and an additional explanation of degrees, radians, and revolution, see Appendix D, Part 5.

Angular Velocity

The rate of rotary displacement is called angular velocity, symbolized as ω (omega). Angular velocity is equal to the angle through which the radius turns divided by the time it takes for the displacement:

It is expressed as degrees/second, radians/second, or revolutions/second. A softball pitcher who moves the arm through an arc of 140 degrees in 0.1 second has an average angular velocity of 1400 degrees per second. This could also be expressed as 3.88 revolutions per second or 24.43 radians per second. This velocity is called average velocity because film studies of pitchers show that the angular displacement during the execution of the skill is not uniform, and a velocity such as this represents the average velocity over the time span through which the displacement is measured. As with linear movements, most angular human movements are likely to be variable and not uniform. The longer the time span through which the displacement is measured, the more variability is averaged. Thus, if one is interested in the velocity at a specific instant in a skill, the displacement must be measured over an extremely small time span.

Figure 11.16 shows the variations in displacement during the execution of a golf drive. Each displacement between images occurs over the same time period (approximately .0067 seconds), so greater spacing between images indicates greater angular displacement over the same time or greater velocity. When the total angular displacement of the golf club was measured from the beginning of the downswing to the point of contact with the ball and divided by the time it took for the swing, the average velocity of the club was 2148 degrees/second (37.5 rad/sec). The “instant” velocity at a, however, was 1432 degrees/sec, and at b it was 2864 degrees/sec. Illustrations such as this, which show a motion as it occurs over very small spans of time, are produced with high-speed film or video. This golfer was filmed at a rate of approximately 150 frames per second, or .0067 seconds per picture.

Angular Acceleration

In the discussion of linear velocity, a change in velocity was called acceleration. The same is true for changes in angular velocity. Angular acceleration α (alpha) is the rate of change of angular velocity and is expressed in equation form as

where ωf is final angular velocity, ωi is initial velocity, and t is time. If, in Figure 11.16, the angular velocity is 25 rad/sec at a and 50 rad/sec at b and the time lapse between a and b is .11 seconds, the angular acceleration between a and b is 241 rad/sec/sec. This value for α, indicating that the velocity increased 241 radians per second each second, would be true, of course, only if the velocity increased at a uniform rate. Otherwise this value has to be considered as an average of accelerations that may have been higher or lower during the time period studied.

Relationship between Linear and Angular Motion

The description of angular motion in terms of displacement, velocity, and acceleration can tell us a great deal about human movements, but nothing in such a description accounts for or shows the effect of the length of the radius on the outcomes of the movements. We know that, all other things being equal, a baseball hit in the middle of a bat will not go as far as one hit at the end, that a ball hit by a tennis racket will travel farther than a ball hit with the hand, and that a golf driver will cause a struck ball to travel farther than a nine iron. In each instance, greater force is imparted to the struck object because the radius of the striking implement (distance between axis and point of contact) is longer and greater linear velocity is generated at its end.

As can be seen in Figure 11.17, lever PA is shorter than lever PB, and lever PB is shorter than lever PC. If all three levers move through the same angular distance in the same amount of time, it is apparent that point C traveled farther in a curvilinear manner than either point A or point B. This curvilinear distance is difficult to measure but may be easily calculated if the angular displacement (θ) and the length of the radius (r) are known. The equation that expresses the relationship between angular and linear displacement is s = θr. Because the linear displacement of point C took the same amount of time as the displacements of points A and B, point C moved with a greater linear velocity than either of the other two points. Point B had a smaller linear velocity than point C but a larger linear velocity than point A. All three levers have the same angular velocity, but linear velocity of the circular motion at the end of each lever is proportional to the length of the lever. An object moved at the end of a long radius will have a greater linear velocity than one moved at the end of a short radius, if the angular velocity is kept constant. The longer the radius, the greater is the linear velocity of a point at the end of that radius. Thus it is to the advantage of a performer to use as long a lever as possible to impart linear velocity to an object if the long lever length does not cause too great a sacrifice in angular velocity. The longer the lever, the more effort it takes to swing it. Therefore, the optimum length of the lever for a person depends on the individual’s ability to maintain angular velocity. A child who cannot handle the weight of a long radius is better off with a shortened implement that can be controlled and swung rapidly, whereas a strong adult profits by using a longer radius.

If the reverse occurs—that is, if the linear velocity is kept constant—an increase in radius will result in a decrease in angular velocity. Once an object is engaged in rotary motion, the linear velocity at the end of the radius stays the same because of the conservation of momentum. The radius of rotation for a pike somersault dive is longer than that for a tuck somersault, and the radius for a layout somersault is longer than that for a pike somersault. If one starts a dive in an open position and then tucks tightly, the radius of rotation decreases but, because the linear velocity does not change, the angular velocity increases. The same situation occurs when a figure skater rotates slowly about a vertical axis with arms and one leg out to the side and brings the arms and leg close to the axis. The radius decreases and the angular velocity increases. To slow down, the skater again reaches out with arms and leg. Figure 11.18 shows the effect of shortening the radius while maintaining a constant linear velocity at the end of the radius. Shortening the radius will increase the angular velocity, and lengthening it will decrease the angular velocity. Points a and b on radii A and B have moved through the same linear distance, but the angular displacement for A is greater than that for B. If the displacements of a and b each take place in the same amount of time, the linear velocities will be equal, but the angular velocity for A will be greater than that for B.

The relationship that exists between the angular velocity of an object moving in a rotary fashion and the linear velocity at the end of its radius is expressed by the equation

To use any of the equations relating linear and angular motion, the angular measures must be expressed in radians. If the angle is expressed in degrees, it can be converted to radians simply by dividing by 57.3.

Either form of the equation shows the direct proportionality that exists between linear velocity and the radius. For any given angular velocity, the linear velocity is proportional to the radius. If the radius doubles, the linear velocity does likewise. And for any given linear velocity, the angular velocity is inversely proportional to the radius. If the radius doubles, the angular velocity decreases by half. To achieve higher linear velocities at the end of levers, the motions must be done with longer levers or higher angular velocities (Figure 11.19).

• 1. Define the following key terms:
  • Linear motion
  • Angular or rotary motion
  • Distance
  • Speed
  • Displacement
  • Velocity
  • Acceleration
  • Gravity
  • Projectile
• 2. Choose one of the following activities:
  • Running long jump
  • Basketball jump shot
  • Softball pitching
  • Identify the critical linear and angular kinematic elements of the selected skill. Explain how each of these elements contributes to successful performance.
• 3. An Olympic skater who participated in the men’s speed-skating events had the following times: 1500 m in 2 min 2.96 sec; 5000 m in 7 min 23.61 sec; and 10,000 m in 15 min 1.35 sec. What was his average speed for each of these events?
• See Appendix H, Chapter 11.
• 4. Using the concept of acceleration, explain how a swimmer can have a better time for a 100-m race in a 25-m pool than in a 100-m pool.
• 5. How much time will a batter have to decide to swing at a pitch and still hit it under these circumstances?
  • a. The pitcher throws the ball at 80 mph.
  • b. The distance from the ball release to the plate is 56 ft.
  • c. It takes the batter .30 sec to get the bat to the desired contact point.
• See Appendix H, Chapter 11.
• 6. With the help of several classmates, prepare a displacement–time graph and a velocity–time graph for your performance on two 50-m dash efforts: one in which you run through the end, and one in which you stop right at 50 m. Class members should be spaced at 5-m intervals along your running path, each with a stopwatch. On the signal for you to go, each timer will start the watch and stop it when you pass that timer’s position. Prepare a table with the following data for each run:
  • a. Distance intervals
  • b. Times recorded at each interval
  • c. Times over each 5-m interval (subtract adjacent times)
  • d. Average velocity over each 5-m interval
  • For each set of data, prepare a displacement–time graph and a velocity–time graph for the whole run. Describe your run in terms of displacement, velocity, and acceleration. Compare your graphs to those of other groups and note any differences.
• 7. An arrow shot straight up into the air reached a height of 75 m. With what velocity did it leave the bow? How long was the arrow in the air?
• See Appendix H, Chapter 11.
• 8. Place one coin near the edge of a table and another on the end of a ruler (as shown in Figure 11.20). While pressing the center of the ruler to the table with an index finger strike one end of it in the direction indicated so that both coins land on the ground. Diagram the path of each. Which hits the floor first? Explain.

• 9. Throw a ball so that it is projected vertically upward. Catch it at the same height it was released. Have a partner measure the time the ball is in the air—that is, from the time of release to the time the ball lands in your hand. Determine the velocity of the ball at the moment of release and the distance the ball traveled before it started its descent. Graph the flight of the ball on a piece of graph paper. Break the flight of the ball into five sections: (1) from initiation of upward motion of the hand to just prior to release, (2) from moment of release to just before maximum height, (3) at maximum height, (4) from maximum height to just before you catch the ball, (5) from the ball contact to the ball being brought to rest. For each section, describe the displacement, velocity, and acceleration.
• 10. While walking along at a constant speed, project a ball vertically into the air. If you continue to walk without changing your speed or direction, where will the ball land? Explain. Draw a diagram of the ball’s flight indicating the forces acting on it.
• 11. Assume that you are able to throw a ball with a velocity of 24 m/sec and at an angle of 45 degrees with the horizontal. If it is caught at the same height from the ground at which it was released, neglecting air resistance, how far will it go? How long will it be in flight? Repeat with a 30-degree angle of release. How would these values change if the landing height were lowered?
• See Appendix H, Chapter 11.
• 12. Using Figure 11.3, determine the angular velocity of the lower leg at the knee joint at the beginning of the force phase and at the moment of foot contact with the ball. The time between each stick figure tracing is .0156 second. What is the linear velocity at the ankle at the moment of contact if the lower leg is 35 centimeters (knee joint to ankle joint)?
• See Appendix H, Chapter 11.