- Discuss the interrelationships among kinematic variables.
- Correctly associate linear kinematic quantities with their units of measure.
- Identify and describe the effects of factors governing projectile trajectory.
- Explain why the horizontal and vertical components of projectile motion are analyzed separately.
- Distinguish between average and instantaneous quantities, and identify the circumstances under which each is a quantity of interest.
- Select and use appropriate equations to solve problems related to linear kinematics.

Why is a sprinter’s acceleration close to zero in the middle of a race? How does the size of a dancer’s foot affect the performance time that a choreographer must allocate for jumps? At what angle should a discus or a javelin be thrown to achieve maximum distance? Why does a ball thrown horizontally hit the ground at the same time as a ball dropped from the same height? These questions all relate to the kinematic characteristics of a pure form of movement: linear motion. This chapter introduces the study of human movement mechanics with a discussion of linear kinematic quantities and projectile motion.

**Kinematics** is the geometry, pattern,
or form of motion with respect to time. Kinematics, which describes
the appearance of motion, is distinguished from kinetics, the forces
associated with motion. Linear kinematics involves the shape, form,
pattern, and sequencing of linear movement through time, without
particular reference to the forces that cause or result from the
motion.

Careful kinematic analyses of performance are invaluable for clinicians, physical activity teachers, and coaches. When people learn a new motor skill, a progressive modification of movement kinematics reflects the learning process. This is particularly true for young children, whose movement kinematics changes with the normal changes in anthropometry and neuromuscular coordination that accompany growth. Likewise, when a patient rehabilitates an injured joint, the therapist or clinician looks for the gradual return of normal joint kinematics.

Kinematics spans both qualitative and quantitative forms of analysis. For example, qualitatively describing the kinematics of a soccer kick entails identifying the major joint actions, including hip flexion, knee extension, and possibly plantar flexion at the ankle. A more detailed qualitative kinematic analysis might also describe the precise sequencing and timing of body segment movements, which translates to the degree of skill evident on the part of the kicker. Although most assessments of human movement are carried out qualitatively through visual observation, quantitative analysis is also sometimes appropriate. Physical therapists, for example, often measure the range of motion of an injured joint to help determine the extent to which range of motion exercises may be needed. When a coach measures an athlete’s performance in the shot put or long jump, this too is a quantitative assessment.

Sport biomechanists often quantitatively study the kinematic factors that characterize an elite performance or the biomechanical factors that may limit the performance of a particular athlete. Sometimes this type of analysis involves construction of a model that details the kinematic characteristics of sound performance for practical use by coaches and athletes. Kinematic analysis can also help to identify injury potential during sport skill execution. For example, a study comparing the kinematics of American and Korean professional baseball pitchers identified a variety of specific differences that contribute to 10% greater ball velocity for the American pitchers but that also likely contribute to increased incidence of injury to the elbow and shoulder (10).

Most biomechanical studies of human kinematics, however, are performed on non-elite subjects. Kinematic research has shown that infants begin to use stable patterns of coordination in reaching for objects at age 12–15 months, with adultlike reaching movements occurring by about 2 years (18). Scientists have also studied the kinematic characteristics of progressive gait development and throwing ability in young children (15). In collaboration with adapted physical education specialists, biomechanists have documented the characteristic kinematic patterns associated with relatively common disabling conditions such as cerebral palsy, Down syndrome, and stroke. Quantitative kinematic screening tests are used to evaluate treatment and progression of a wide variety of motor disorders (30). Kinematic analysis indicates that walking on a treadmill with a grade just greater than 12% may be optimal for minimizing patellofemoral discomfort and potential strain on the anterior cruciate ligament (ACL) in post–ACL reconstruction patients (20).

Biomechanists commonly use high-speed cinematography or videography to perform quantitativekinematic analyses. The process involves taking a carefully planned film or video of a performance, with subsequent computerized or computer-assisted analysis of the performance on a picture-by-picture basis, as described in Chapter 2.

Units of distance and displacement are units of length. In the
metric system, the most commonly used unit of distance and displacement
is the **meter** (m). A kilometer (km)
is 1000 m, a centimeter (cm) is 1/100
m, and a millimeter (mm) is 1/1000
m. In the English system, common units of length are the inch,
the foot (0.30 m), the yard (0.91 m), and the mile (1.61 km).

Distance and displacement are assessed differently. Distance
is measured along the path of motion. When a runner completes 11/2 laps
around a 400 m track, the distance that the runner has covered is
equal to 600 (400 + 200) m. **Linear
displacement** is measured in a straight line from position 1
to position 2, or from initial position to final position. At the
end of 11/2 laps around the track, the runner’s
displacement is the length of the straight imaginary line that transverses
the field, connecting the runner’s initial position to
the runner’s final position halfway around the track (see Introductory
Problem 1). At the completion of 2 laps around the track, the
distance run is 800 m. Because initial and final positions are the
same, however, the runner’s displacement is zero. When
a skater moves around a rink, the distance the skater travels may
be measured along the tracks left by the skates. The skater’s
displacement is measured along a straight line from initial to final
positions on the ice (Figure 10-1).

###### Figure 10-1

The distance a skater travels may be measured from the track on the ice. The skater’s displacement is measured in a straight line from initial position to final position.

Another difference is that distance is a scalar quantity while
displacement is a vector quantity. Consequently, the displacement
includes more than just the length of the line between two positions. Of
equal importance is the *direction *in
which the displacement occurs. The direction of a displacement relates
the final position to the initial position. For example, the displacement
of a yacht that has sailed 900 m on a tack due south would be identified
as 900 m to the south.

The direction of a displacement may be indicated in several different,
equally acceptable ways. Compass directions such as south and northwest
and the terms *left/right, up/down,* and *positive/negative* are all
appropriate labels. The positive direction is typically defined
as upwards and/or to the right, with negative regarded
as downwards and/or to the left. This enables indication
of direction using plus and minus signs. The most important thing
is to be consistent in using the system or convention adopted for
indicating direction in a given context. It would be confusing to
describe a displacement as 500 m north followed by 300 m to the
right.

Either distance or displacement may be the more important quantity of interest depending on the situation. Many 5 km and 10 km racecourses are set up so that the finish line is only a block or two from the starting line. Participants in these races are usually interested in the number of kilometers of distance covered or the number of kilometers left to cover as they progress along the racecourse. Knowledge of displacement is not particularly valuable during this type of event. In other situations, however, displacement is more important. For example, triathlon competitions may involve a swim across a lake. Because swimming in a perfectly straight line across a lake is virtually impossible, the actual distance a swimmer covers is always somewhat greater than the width of the lake (see Sample Problem 10.1). However, the course is set up so that the identified length of the swim course is the length of the displacement between the entry and exit points on the lake.

Displacement magnitude and distance covered can be identical. When a cross-country skier travels down a straight path through the woods, both distance covered and displacement are equal. However, any time the path of motion is not rectilinear, the distance traveled and the size of the displacement will differ.

A swimmer crosses a lake that is 0.9 km wide in 30 minutes. What was his average velocity? Can his average speed be calculated?

After reading the problem carefully, the next step is to sketch the problem situation, showing all quantities that are known or may be deduced from the problem statement:

In this situation, we know that the swimmer’s displacement is 0.9 km. However, we know nothing about the exact path that he may have followed. The next step is to identify the appropriate formula to use to find the unknown quantity, which is velocity:

The known quantities can now be filled in to solve for velocity:

Speed is calculated as distance divided by time. Although we know the time taken to cross the lake, we do not know, nor can we surmise from the information given, the exact distance covered by the swimmer. Therefore, his speed cannot be calculated.

Two quantities that parallel distance and linear displacement
are speed and **linear velocity**. These
terms are often used synonymously in general conversation, but in
mechanics, they have precise and different meanings. *Speed,* a scalar quantity, is defined
as the distance covered divided by the time taken to cover it:

*Velocity* (v) is the change in position,
or displacement, that occurs during a given period of time:

Because the Greek capital letter delta (Δ) is commonly used in mathematical expressions to mean “change in,” a shorthand version of the relationship expressed follows, with t representing the amount of time elapsed during the velocity assessment:

Another way to express change in position is position2 − position1, in which position1 represents the body’s position at one point in time and position2 represents the body’s position at a later point:

Because velocity is based on displacement, it is also a vector quantity. Consequently, description of velocity must include an indication of both the direction and the magnitude of the motion. If the direction of the motion is positive, velocity is positive; if the direction is negative, velocity is negative. A change in a body’s velocity may represent a change in its speed, movement direction, or both.

Whenever two or more velocities act, the laws of vector algebra govern the ultimate speed and direction of the resultant motion. For example, the path actually taken by a swimmer crossing a river is determined by the vector sum of the swimmer’s speed in the intended direction and the velocity of the river’s current (Figure 10-2). Sample Problem 10.2 provides an illustration of this situation.

Units of speed and velocity are units of length divided by units of time. In the metric system, common units for speed and velocity are meters per second (m/s) and kilometers per hour (km/hr). However, any unit of length divided by any unit of time yields an acceptable unit of speed or velocity. For example, a speed of 5 m/s can also be expressed as 5000 mm/s or 18,000 m/hr. It is usually most practical to select units that will result in expression of the quantity in the smallest, most manageable form.

For human gait, speed is the product of stride length and stride frequency. Adults in a hurry tend to walk with both longer stride lengths and faster stride frequency than they use under more leisurely circumstances. However, for toddlers, women wearing high-heeled shoes, and elderly individuals, it is often difficult to significantly increase stride length without losing balance. Stride lengths during normal walking gait among young children are characterized by large variability, with stride consistency gradually increasing through the early teenage years (13).

Increased variability of walking kinematics has also been associated with elevated risk of falls in older adults. Assessing kinematic variability is not an easy task, however. According to one research study, accurate determination of the kinematic variation in human gait requires analysis of at least 400 steps (28)! A large degree of variability in gait kinematics is apparently present in other species as well. Researchers examining the trotting gait of 10 Labrador retrievers on a treadmill found that the dogs’ gait was not repeatable either for individual dogs at different points in the data collection or between different dogs at the same point in the data collection (6).

During running, a kinematic variable such as stride length is not simply a function of the runner’s body height, but is also influenced by muscle fiber composition, footwear, level of fatigue, injury history, and the inclination (grade) and stiffness of the running surface (2, 12). Runners traveling at a slow pace tend to increase velocity primarily by increasing stride length. At faster running speeds, recreational runners rely more on increasing stride frequency to increase velocity (Figure 10-3). In cross-country skiing, as speed increases, stride rate increases and stride length tends to decrease (27). Overstriding, or using an overly long stride length, should be avoided in both running and skiing, since it is a risk factor for hamstring strains.

Those who run regularly for exercise usually prefer a given stride frequency over a range of slow-to-moderate running speeds. One reason for this may be related to running economy—the oxygen consumption required for performing a given task (1). Most runners tend to choose a combination of stride length and stride frequency that minimizes the physiological cost of running (3). As discussed in Chapter 1, many species of animals do the same thing. Running on downhill and uphill surfaces tends to respectively increase and decrease running speed, with these differences primarily a function of increased and decreased stride length (29). The presence of fatigue, as would be expected near the end of a marathon event, tends to result in increased stride frequency and decreased stride length (19).

Since maximizing speed is the objective of all racing events, sport biomechanists have focused on the kinematic features that appear to accompany fast performances in running, skiing, skating, cycling, swimming, and rowing events. Research has shown that the best male and female sprinters are distinguished from their less-skilled counterparts by extremely high stride frequencies and short ground contact times, although their stride lengths are usually only average or slightly greater than average (9). In contrast, the fastest cross-country skiers have longer-than-average cycle lengths, with cycle rates that are only average (32). Research on skating kinematics has shown that better ice skaters appear to excel because of higher stride rates (23), whereas elite roller skaters are distinguished by longer strides (37). Interestingly, analysis of elite swimmers in the 100 m and 200 m breaststroke events has demonstrated that the optimum ratio of stroke rate to stroke length is unique to each individual swimmer (34).

When racing performances are analyzed, comparisons are usually
based on pace rather than speed or velocity. *Pace* is
the inverse of speed. Rather than units of distance divided by units
of time, pace is presented as units of time divided by units of
distance. Pace is the time taken to cover a given distance and is commonly
quantified as minutes per km or minutes per mile.

A swimmer orients herself perpendicular to the parallel banks of a river. If the swimmer’s velocity is 2 m/s and the velocity of the current is 0.5 m/s, what will be her resultant velocity? How far will she actually have to swim to get to the other side if the banks of the river are 50 m apart?

A diagram showing vector representations of the velocities of the swimmer and the current is drawn:

The resultant velocity can be found graphically by measuring the length and the orientation of the vector resultant of the two given velocities:

The resultant velocity can also be found using trigonometric relationships. The magnitude of the resultant velocity may be calculated using the Pythagorean theorem:

The direction of the resultant velocity may be calculated using the cosine relationship:

If the swimmer travels in a straight line in the direction of her resultant velocity, the cosine relationship may be used to calculate her resultant displacement:

We are well aware that the consequence of pressing down or letting
up on the accelerator (gas) pedal of an automobile is usually a
change in the automobile’s speed (and velocity). **Linear acceleration** (a) is defined
as the rate of change in velocity, or the change in velocity occurring
over a given time interval (t):

Another way to express change in velocity is v2 − v1, where v1 represents velocity at one point in time and v2 represents velocity at a later point:

Units of acceleration are units of velocity divided by units of time. If a car increases its velocity by 1 km/hr each second, its acceleration is 1 km/hr/s. If a skier increases velocity by 1 m/s each second, the acceleration is 1 m/s/s. In mathematical terms, it is simpler to express the skier’s acceleration as 1 m/s squared (1 m/s2). A common unit of acceleration in the metric system is m/s2.

Acceleration is the rate of change in velocity, or the degree with which velocity is changing with respect to time. For example, a body accelerating in a positive direction at a constant rate of 2 m/s2 is increasing its velocity by 2 m/s each second. If the body’s initial velocity was zero, a second later its velocity would be 2 m/s, a second after that its velocity would be 4 m/s, and a second after that its velocity would be 6 m/s.

In general usage, the term *accelerating* means
speeding up, or increasing in velocity. If v2 is greater
than v1, acceleration is a positive number, and the body
in motion may have speeded up during the period in question. However,
because it is sometimes appropriate to label the direction of motion
as positive or negative, a positive value of acceleration may not
mean that the body is speeding up.

If the direction of motion is described in terms other than positive or negative, a positive value of acceleration does indicate that the body being analyzed has speeded up. For example, if a sprinter’s velocity is 3 m/s on leaving the blocks and is 5 m/s a second later, calculation of the acceleration that has occurred will yield a positive number. Because v1 = 3 m/s, v2 = 5 m/s, and t = 1 s:

Whenever the direction of motion is described in terms other than positive or negative, and v2 is greater than v1, the value of acceleration will be a positive number, and the object in question is speeding up.

Acceleration can also assume a negative value. As long as the direction of motion is described in terms other than positive or negative, negative acceleration indicates that the body in motion is slowing down, or that its velocity is decreasing. For example, when a base runner slides to a stop over home plate, acceleration is negative. If a base runner’s velocity is 4 m/s when going into a 0.5 s slide that stops the motion, v1 = 4 m/s, v2 = 0, and t = 0.5 s. Acceleration may be calculated as the following:

Whenever v1 is greater than v2 in this type of situation, acceleration will be negative. Sample Problem 10.3 provides another example of a situation involving negative acceleration.

Understanding acceleration is more complicated when one direction is designated as positive and the opposite direction is designated as negative. In this situation, a positive value of acceleration can indicate either that the object is speeding up in a positive direction or that it is slowing down in a negative direction (Figure 10-4).

Consider the case of a ball being dropped from a hand. As the ball falls faster and faster because of the influence of gravity, it is gaining speed—for example, 0.3 m/s to 0.5 m/s to 0.8 m/s. Because the downward direction is considered as the negative direction, the ball’s velocity is actually –0.3 m/s to –0.5 m/s to –0.8 m/s. If v1 = –0.3 m/s, v2 = –0.5 m/s, and t = 0.02 s, acceleration is calculated as follows:

In this situation, the ball is speeding up, yet its acceleration is negative because it is speeding up in a negative direction. If acceleration is negative, velocity may be either increasing in a negative direction or decreasing in a positive direction. Alternatively, if acceleration is positive, velocity may be either increasing in a positive direction or decreasing in a negative direction.

The third alternative is for acceleration to be equal to zero. Acceleration is zero whenever velocity is constant, that is, when v1 and v2 are the same. In the middle of a 100 m sprint, a sprinter’s acceleration should be close to zero, because at that point the runner should be running at a constant, near-maximum velocity.

Acceleration and deceleration (the lay term for negative acceleration) have implications for injury of the human body, since changing velocity results from the application of force (see Chapter 12). The anterior cruciate ligament, which restricts the forward sliding of the femur on the tibial plateaus during knee flexion, is often injured when an athlete who is running decelerates rapidly or changes directions quickly.

It is important to remember that since acceleration is a vector quantity, changing directions, even while maintaining a constant speed, represents a change in acceleration. The concept of angular acceleration, with direction constantly changing, is discussed in Chapter 11. The forces associated with change in acceleration based on change in direction must be compensated for by skiers and velodrome cyclists, in particular. That topic is discussed in Chapter 14.

A soccer ball is rolling down a field. At t = 0, the ball has an instantaneous velocity of 4 m/s. If the acceleration of the ball is constant at −0.3 m/s2, how long will it take the ball to come to a complete stop?

After reading the problem carefully, the next step is to sketch the problem situation, showing all quantities that are known or given in the problem statement.

The next step is to identify the appropriate formula to use to find the unknown quantity:

The known quantities can now be filled in to solve for the unknown variable (time):

Rearranging the equation, we have the following:

Simplifying the expression on the right side of the equation, we have the solution:

It is often of interest to determine the velocity of acceleration
of an object or body segment at a particular time. For example,
the **instantaneous**velocity of a shot
or a discus at the moment the athlete releases it greatly affects
the distance that the implement will travel. It is sometimes sufficient
to quantify the **average** speed or velocity
of the entire performance.

When speed and velocity are calculated, the procedures depend on whether the average or the instantaneous value is the quantity of interest. Average velocity is calculated as the final displacement divided by the total time. Average acceleration is calculated as the difference in the final and initial velocities divided by the entire time interval. Calculation of instantaneous values can be approximated by dividing differences in velocities over an extremely small time interval. With calculus, velocity can be calculated as the derivative of displacement, and acceleration as the derivative of velocity.

Selection of the time interval over which speed or velocity is quantified is important when analyzing the performance of athletes in racing events. Many athletes can maintain world record paces for the first one-half or three-fourths of the event, but slow during the last leg because of fatigue. In a study involving female high school sprinters performing the 100 m dash, it was found that maximum running speeds of 8.0–8.4 m/s were reached 23–37 m from the start, and that an average of 7.3% of maximum speed was lost when the runners entered the final 10 m (4). Alternatively, some athletes may intentionally perform at a controlled pace during earlier segments of a race and then achieve maximum speed at the end. The longer the event is, the more information is potentially lost or concealed when only the final time or average speed is reported.

Bodies projected into the air are **projectiles**.
A basketball, a discus, a high jumper, and a sky diver are all projectiles
as long as they are moving through the air unassisted. Depending
on the projectile, different kinematic quantities are of interest.
The resultant horizontal displacement of the projectile determines
the winner of the contest in field events such as the shot put,
discus throw, and javelin throw. High jumpers and pole-vaulters maximize
ultimate vertical displacement to win events. Sky divers manipulate
both horizontal and vertical components of velocity to land as close
as possible to targets on the ground.

However, not all objects that fly through the air are projectiles. A projectile is a body in free fall that is subject only to the forces of gravity and air resistance. Therefore, objects such as airplanes and rockets do not qualify as projectiles, because they are also influenced by the forces generated by their engines.

Just as it is more convenient to analyze general motion in terms of its linear and angular components, it is usually more meaningful to analyze the horizontal and vertical components of projectile motion separately. This is true for two reasons. First, the vertical component is influenced by gravity, whereas no force (neglecting air resistance) affects the horizontal component. Second, the horizontal component of motion relates to the distance the projectile travels, and the vertical component relates to the maximum height achieved by the projectile. Once a body has been projected into the air, its overall (resultant) velocity is constantly changing because of the forces acting on it. When examined separately, however, the horizontal and vertical components of projectile velocity change predictably.

Horizontal and vertical components of projectile motion are independent of each other. In the example shown in Figure 10-5, a baseball is dropped from a height of 1 m at the same instant that a second ball is horizontally struck by a bat at a height of 1 m, resulting in a line drive. Both balls land on the level field simultaneously, because the vertical components of their motions are identical. However, because the line drive also has a horizontal component of motion, it undergoes some horizontal displacement as well.

A major factor that influences the vertical but not the horizontal component of projectile motion is the force of gravity, which accelerates bodies in a vertical direction toward the surface of the earth (Figure 10-6). Unlike aerodynamic factors that may vary with the velocity of the wind, gravitational force is a constant, unchanging force that produces a constant downward vertical acceleration. Using the convention that upward is positive and downward is negative, the acceleration of gravity is treated as a negative quantity (−9.81 m/s2). This acceleration remains constant regardless of the size, shape, or weight of the projectile. The vertical component of the initial projection velocity determines the maximum vertical displacement achieved by a body projected from a given relative projection height.

Figure 10-7 illustrates the influence of gravity on
projectile flight in the case of a ball tossed into the air by a juggler.
The ball leaves the juggler’s hand with a certain vertical
velocity. As the ball travels higher and higher, the magnitude of
its velocity decreases because it is undergoing a negative acceleration (the
acceleration of gravity in a downward direction). At the peak or **apex** of the flight, which is that
instant between going up and coming down, vertical velocity is zero. As
the ball falls downward, its speed progressively increases, again
because of gravitational acceleration. Since the direction of motion
is downward, the ball’s velocity is becoming progressively more
negative. If the ball is caught at the same height from which it
was tossed, the ball’s speed is exactly the same as its
initial speed, although its direction is now reversed. Graphs of
the vertical displacement, velocity, and acceleration of a tossed
ball are shown in Figure 10-8.

If an object were projected in a vacuum (with no air resistance), the horizontal component of its velocity would remain exactly the same throughout the flight. However, in most real-life situations, air resistance affects the horizontal component of projectile velocity. A ball thrown with a given initial velocity in an outdoor area will travel much farther if it is thrown with a tailwind rather than into a headwind. Because the effects of air resistance are variable, however, for purposes of simplification, the horizontal component of a given projectile’s velocity will be regarded as an unchanging (constant) quantity in this chapter.

When a projectile drops vertically through the air in a typical reallife situation, its velocity at any point is also related to air resistance. A sky diver’s velocity, for example, is much smaller after the opening of the parachute than before its opening.

Three factors influence the **trajectory** (flight
path) of a projectile: (a) the angle of projection, (b) the projection
speed, and (c) the relative height of projection (Figure 10-9)
(Table 10-1). Understanding how these factors interact
is useful within the context of sport both for determining how to
best project balls and other implements and for predicting how to
best catch or strike projected balls.

Variable | Factors of Influence |
---|---|

Flight time | Initial vertical velocity Relative projection height |

Horizontal displacement | Horizontal velocity Relative projection height |

Vertical displacement | Initial vertical velocity Relative projection height |

Trajectory | Initial speed Projection angle Relative projection height |

The **angle of projection**and the
effects of air resistance govern the shape of a projectile’s
trajectory. Changes in projection speed influence the size of the
trajectory, but trajectory shape is solely dependent on projection
angle. In the absence of air resistance, the trajectory of a projectile
assumes one of three general shapes, depending on the angle of projection.
If the projection angle is perfectly vertical, the trajectory is
also perfectly vertical, with the projectile following the same
path straight up and then straight down again. If the projection
angle is oblique (at some angle between 0˚ and 90˚),
the trajectory is *parabolic,* or shaped
like a parabola. A parabola is symmetrical, so its right and left
halves are mirror images of each other. A body projected perfectly
horizontally (at an angle of 0˚) will follow a trajectory resembling
one-half of a parabola (Figure 10-10). Figure 10-11 displays
scaled, theoretical trajectories for an object projected at different
angles at a given speed. A ball thrown upward at a projection angle
of 80˚ to the horizontal follows a relatively high and narrow
trajectory, achieving more height than horizontal distance. A ball
projected upward at a 10˚ angle to the horizontal follows
a trajectory that is flat and long in shape.

Projection angle has direct implications for success in the sport of basketball, since a steep angle of entry into the basket allows a somewhat larger margin of error than a shallow angle of entry. Within 4.57 m of the basket, jump shot release angles are about 52–55˚, providing a relatively steep angle of entry, whereas shots taken from 6.40 m tend to be released at 48–50˚, allowing for a minimum release speed, but a less steep angle of entry (25). When shooting in close proximity to a defender, players tend to release the ball at a greater release angle and from a greater height than is the case when a player is open (31). Although the strategy behind this is typically to keep the shot from being blocked, it may also result in more accurate shooting.

In projection situations on a field, air resistance may, in reality, create irregularities in the shape of a projectile’s trajectory. A typical modification in trajectory caused by air resistance is displayed in Figure 10-12. For purposes of simplification, the effects of aerodynamic forces will be disregarded in the discussion of projectile motion.

When projection angle and other factors are constant, the **projection speed** determines the length
or size of a projectile’s trajectory. For example, when
a body is projected vertically upward, the projectile’s
initial speed determines the height of the trajectory’s
apex. For a body that is projected at an oblique angle, the speed
of projection determines both the height and the horizontal length
of the trajectory (Figure 10-13). The combined effects
of projection speed and projection angle on the horizontal displacement,
or **range**, of a projectile are shown
in Table 10-2.

Projection Speed (m/s) | Projection Angle (˚) | Range (m) |
---|---|---|

10 | 10 | 3.49 |

10 | 20 | 6.55 |

10 | 30 | 8.83 |

10 | 40 | 10.04 |

10 | 45 | 10.19 |

10 | 50 | 10.04 |

10 | 60 | 8.83 |

10 | 70 | 6.55 |

10 | 80 | 3.49 |

20 | 10 | 13.94 |

20 | 20 | 26.21 |

20 | 30 | 35.31 |

20 | 40 | 40.15 |

20 | 45 | 40.77 |

20 | 50 | 40.15 |

20 | 60 | 35.31 |

20 | 70 | 26.21 |

20 | 80 | 13.94 |

30 | 10 | 31.38 |

30 | 20 | 58.97 |

30 | 30 | 79.45 |

30 | 40 | 90.35 |

30 | 45 | 91.74 |

30 | 50 | 90.35 |

30 | 60 | 79.45 |

30 | 70 | 58.97 |

30 | 80 | 31.38 |

Performance in the execution of a vertical jump on a flat surface is entirely dependent on takeoff speed; that is, the greater the vertical velocity at takeoff, the higher the jump, and the higher the jump, the greater the amount of time the jumper is airborne. Elite beach volleyball players can jump higher and stay airborne longer when taking off from a solid surface than from sand because the instability of the sand produces a reduction in takeoff velocity (11).

The time required for the performance of a vertical jump can be an important issue for dance choreographers. The incorporation of vertical jumps into a performance must be planned carefully (21). If the tempo of the music necessitates that vertical jumps be executed within one-third of a second, the height of the jumps is restricted to approximately 12 cm. The choreographer must be aware that under these circumstances, most dancers do not have sufficient floor clearance to point their toes during jump execution.

The third major factor influencing projectiletrajectory is the **relative projection height** (Figure
10-14). This is the difference in the height from which the
body is initially projected and the height at which it lands or
stops. When a discus is released by a thrower from a height of 11/2 m
above the ground, the relative projection height is 11/2 m,
because the projection height is 11/2 m greater
than the height of the field on which the discus lands. If a driven
golf ball becomes lodged in a tree, the relative projection height
is negative, because the landing height is greater than the projection
height. When projection velocity is constant, greater relative projection
height translates to longer flight time and greater horizontal displacement
of the projectile.

In the sport of diving, relative projection height is the height of the springboard or platform above the water. If a diver’s center of gravity is elevated 1.5 m above the springboard at the apex of the trajectory, flight time is about 1.2 s from a 1 m board and 1.4 s from a 3 m board. This provides enough time for a skilled diver to complete 3 somersaults from a 1 m board and 31/2-somersaults from a 3 m board (38). The implication is that a diver attempting to learn a 31/2-somersault dive from the 3 m springboard should first be able to easily execute a 21/2 somersault dive from the 1 m board.

In sporting events based on achieving maximum horizontal displacement or maximum vertical displacement of a projectile, the athlete’s primary goal is to maximize the speed of projection (26). In the throwing events, another objective is to maximize release height, because greater relative projection height produces longer flight time, and consequently greater horizontal displacement of the projectile. However, it is generally not prudent for a thrower to sacrifice release speed for added release height.

The factor that varies the most, with both the event and the performer, is the optimum angle of projection. When relative projection height is zero, the angle of projection that produces maximum horizontal displacement is 45˚. As relative projection height increases, the optimum angle of projection decreases, and as relative projection height decreases, the optimum angle increases (Figure 10-15).

###### Figure 10-15

When projection speed is constant and aerodynamics are not considered, the optimum projection angle is based on the relative height of projection. When the relative projection height is zero, an angle of 45˚ is optimum. As the relative projection height increases, optimum projection angle decreases. As the relative projection height becomes increasingly negative, the optimum projection angle increases.

It is important to recognize that there are constraint relationships among projection speed, height, and angle, such that when one is shifted closer to what would theoretically be optimal, another moves farther away from optimum. This is because humans are not machines, and human anatomy dictates certain constraints. Research has shown, for example, that the constraint relationships among release speed, height, and angle for performance in the shot put are such that achievable release speed decreases with increasing release angle at 1.7 (m/s)/rad and decreases with increasing release height at 0.8 (m/s)/m (17).

Likewise, when the human body is the projectile during a jump, high takeoff speed serves to constrain the projection angle that can be achieved (33). In the performance of the long jump, for example, because takeoff and landing heights are the same, the theoretically optimum angle of takeoff is 45˚ with respect to the horizontal. However, it has been estimated by Hay (14) that to obtain this theoretically optimum takeoff angle, long jumpers would decrease the horizontal velocity they could otherwise obtain by approximately 50%. Research has shown that success in the long jump, high jump, and pole vault are all related to the athlete’s ability to maximize horizontal velocity going into takeoff (5, 8, 35). The actual takeoff angles employed by elite long jumpers range from approximately 18˚ to 27˚ (14). Takeoff angles during all three phases of the triple jump are even smaller for elite performers than those used in the long jump (24). Performance in the triple jump is complicated by the fact that there is a direct trade-off between horizontal velocity and vertical velocity during the jumps (39). In the ski jump, where athletes have the advantage of a large relative height between takeoff and landing, takeoff angles are as small as 4.6–6.2˚ (36). In an event such as the high jump, in which the goal is to maximize vertical displacement, takeoff angles among skilled Fosbury Flop–style jumpers range from 40˚–48˚ (7).

In the throwing events, the aerodynamic characteristics of the projected implements also influence the trajectory (see Chapter 15). In these events (shot, discus, javelin, and hammer), only the trajectory of the shot is not appreciably affected by aerodynamic forces. The concept that the optimum angle of release must not restrict release speed is still a paramount consideration for performance in the shot put. The release angles reported among elite shot-putters are approximately 36–37˚ (16). It has been shown, however, that optimum release angle for the shot put varies from athlete to athlete because of individual differences in the decrease of release speed with increasing release angle (22).

Because velocity is a vector quantity, the **initial
velocity** of a projectile incorporates both the initial speed
(magnitude) and the angle of projection (direction) into a single
quantity. When the initial velocity of a projectile is resolved
into horizontal and vertical components, the horizontal component
has a certain speed or magnitude in a horizontal direction, and
the vertical component has a speed or magnitude in a vertical direction
(Figure 10-16). The magnitudes of the horizontal and vertical
components are always quantified so that if they were added together
through the process of vector composition, the resultant velocity
vector would be equal in magnitude and direction to the original
initial velocity vector. The horizontal and vertical components
of initial velocity may be quantified both graphically and trigonometrically
(see Sample Problem 10.4).

For purposes of analyzing the motion of projectiles, it will be assumed that the horizontal component of projectile velocity is constant throughout the trajectory and that the vertical component of projectile velocity is constantly changing because of the influence of gravity (Figure 10-17). Since horizontal projectile velocity is constant, horizontal acceleration is equal to the constant of zero throughout the trajectory. The vertical acceleration of a projectile is equal to the constant 29.81 m/s2.

A basketball is released with an initial speed of 8 m/s at an angle of 60˚. Find the horizontal and vertical components of the ball’s initial velocity, both graphically and trigonometrically.

A diagram showing a vector representation of the initial velocity is drawn using a scale of 1 cm = 2 m/s:

The horizontal component is drawn in along the horizontal line to a length that is equal to the length that the original velocity vector extends in the horizontal direction. The vertical component is then drawn in the same fashion in a direction perpendicular to the horizontal line:

The lengths of the horizontal and vertical components are then measured:

- length of horizontal component = 2 cm
- length of vertical component = 3.5 cm

To calculate the magnitudes of the horizontal and vertical components, use the scale factor of 2 m/s/cm:

Magnitude of horizontal component:

Magnitude of vertical component:

To solve for vh and vv trigonometrically, construct a right triangle with the sides being the horizontal and vertical components of initial velocity and the initial velocity represented as the hypotenuse:

The sine and cosine relationships may be used to quantify the horizontal and vertical components:

Note that the magnitude of the horizontal component is *always* equal to the magnitude of the
initial velocity multiplied by the cosine of the projection angle.
Similarly, the magnitude of the initial vertical component is *always* equal to the magnitude of the
initial velocity multiplied by the sine of the projection angle.

When a body is moving with a constant acceleration (positive,
negative, or equal to zero), certain interrelationships are present
among the kinematic quantities associated with the motion of the body.
These interrelationships may be expressed using three mathematical
equations originally derived by Galileo, which are known as the **laws of constant acceleration**, or
the laws of uniformly accelerated motion. Using the variable symbols
d, v, a, and t (representing displacement, velocity, acceleration,
and time, respectively) and with the subscripts 1 and 2 (representing
first or initial and second or final points in time), the equations
are the following:

Notice that each of the equations contains a unique combination of three of the four kinematic quantities: displacement, velocity, acceleration, and time. This provides considerable flexibility for solving problems in which two of the quantities are known and the objective is to solve for a third. The symbols used in these equations are listed in Table 10-3.

Symbol | Meaning | Representing in Equations |
---|---|---|

d | Displacement | Change in position |

v | Velocity | Rate of change in position |

a | Acceleration | Rate of change in velocity |

t | Time | Time interval |

v1 | Initial or first velocity | Velocity at time 1 |

v2 | Later or final velocity | Velocity at time 2 |

vv | Vertical velocity | Vertical component of total velocity |

vh | Horizontal velocity | Horizontal component of total velocity |

It is instructive to examine these relationships as applied to the horizontal component of projectile motion in which a = 0. In this case, each term containing acceleration may be removed from the equation. The equations then appear as the following:

Equations 1H and 3H reaffirm that the horizontal component of projectile velocity is a constant. Equation 2H indicates that horizontal displacement is equal to the product of horizontal velocity and time (see Sample Problem 10.5).

When the constant acceleration relationships are applied to the vertical component of projectile motion, acceleration is equal to –9.81 m/s2, and the equations cannot be simplified by the deletion of the acceleration term. However, in analysis of the vertical component of projectile motion, the initial velocity (v1) is equal to zero in certain cases. For example, when an object is dropped from a stationary position, the initial velocity of the object is zero. When this is the case, the equations of constant acceleration may be expressed as the following:

When an object is dropped, equation 1V relates that the object’s velocity at any instant is the product of gravitational acceleration and the amount of time the object has been in free fall. Equation 2V indicates that the vertical distance through which the object has fallen can be calculated from gravitational acceleration and the amount of time the object has been falling. Equation 3V expresses the relationship between the object’s velocity and vertical displacement at a certain time and gravitational acceleration.

It is useful in analyzing projectile motion to remember that at the apex of a projectile’s trajectory, the vertical component of velocity is zero. If the goal is to determine the maximum height achieved by a projectile, v2 in equation 3 may be set equal to zero:

An example of this use of equation 3A is shown in Sample Problem 10.6. If the problem is to determine the total flight time, one approach is to calculate the time it takes to reach the apex, which is one-half of the total flight time if the projection and landing heights are equal. In this case, v2 in equation 1 for the vertical component of the motion may be set equal to zero because vertical velocity is zero at the apex:

Sample Problem 10.7 illustrates this use of equation 1A.

When using the equations of constant acceleration, it is important to remember that they may be applied to the horizontal component of projectile motion or to the vertical component of projectile motion, but not to the resultant motion of the projectile. If the horizontal component of motion is being analyzed, a = 0, but if the vertical component is being analyzed, a = −9.81 m/s2. The equations of constant acceleration and their special variations are summarized in Table 10-4.

The Equations of Constant Acceleration |

These equations may be used to relate linearkinematic quantities whenever acceleration (a) is a constant, unchanging value: |

Special Case Applications of the Equations of Constant Acceleration |

For the horizontal component of projectile motion, with a = 0: |

For the vertical component of projectile motion, with v1 = 0, as when the projectile is dropped from a static position: |

For the vertical component of projectile motion, with v2 = 0, as when the projectile is at its apex: |

The score was tied 20–20 in the final 1987 AFC playoff game between the Denver Broncos and the Cleveland Browns. During the first overtime period, Denver had the opportunity to kick a field goal, with the ball placed at a distance of 29 m from the goalposts. If the ball was kicked with the horizontal component of initial velocity being 18 m/s and a flight time of 2 s, was the kick long enough to make the field goal?

vh = 18 m/s

t = 2 s

Equation 2H is selected to solve the problem, since two of the variables contained in the formula (vh and t) are known quantities, and since the unknown variable (d) is the quantity we wish to find:

The ball did travel a sufficient distance for the field goal to be good, and Denver won the game, advancing to Super Bowl XXI.

A volleyball is deflected vertically by a player in a game housed in a high school gymnasium where the ceiling clearance is 10 m. If the initial velocity of the ball is 15 m/s, will the ball contact the ceiling?

v1 = 15 m/s

a = −9.81 m/s2

The equation selected for use in solving this problem must contain the variable d for vertical displacement. Equation 2 contains d but also contains the variable t, which is an unknown quantity in this problem. Equation 3 contains the variable d, and, recalling that vertical velocity is zero at the apex of the trajectory, Equation 3A can be used to find d:

Therefore, the ball has sufficient velocity to contact the 10 m ceiling.

A ball is kicked at a 35˚ angle, with an initial speed of 12 m/s. How high and how far does the ball go?

vh = 12 cos 35 m/s

vv = 12 sin 35 m/s

*How high does the ball go?*

Equation 1 cannot be used because it does not contain d. Equation 2 cannot be used unless t is known. Since vertical velocity is zero at the apex of the ball’s trajectory, equation 3A is selected:

*How far does the ball go?*

Equation 2H for horizontal motion cannot be used because t for which the ball was in the air is not known. Equation 1A can be used to solve for the time it took the ball to reach its apex:

Recalling that the time to reach the apex is one-half of the total flight time, total time is the following:

Equation 2H can then be used to solve for the horizontal distance the ball traveled:

Linearkinematics is the study of the form or sequencing of linear motion with respect to time. Linear kinematic quantities include the scalar quantities of distance and speed, and the vector quantities of displacement, velocity, and acceleration. Depending on the motion being analyzed, either a vector quantity or its scalar equivalent and either an instantaneous or an average quantity may be of interest.

A projectile is a body in free fall that is affected only by gravity and air resistance. Projectile motion is analyzed in terms of its horizontal and vertical components. The two components are independent of each other, and only the vertical component is influenced by gravitational force. Factors that determine the height and distance the projectile achieves are projection angle, projection speed, and relative projection height. The equations of constant acceleration can be used to quantitatively analyze projectile motion, with vertical acceleration being 29.81 m/s2 and horizontal acceleration being zero.

1. A runner completes 61/2 laps around a 400 m track during a 12 min (720 s) run test. Calculate the following quantities:

a. The distance the runner covered

b. The runner’s displacement at the end of 12 min

c. The runner’s average speed

d. The runner’s average velocity

e. The runner’s average pace

(Answers: a. 2.6 km; b. 160 m; c. 3.6 m/s; d. 0.22 m/s; e. 4.6 min/km)

2. A ball rolls with an acceleration of 20.5 m/s2. If it stops after 7 s, what was its initial speed? (Answer: 3.5 m/s)

3. A wheelchair marathoner has a speed of 5 m/s after rolling down a small hill in 1.5 s. If the wheelchair underwent a constant acceleration of 3 m/s2 during the descent, what was the marathoner’s speed at the top of the hill? (Answer: 0.5 m/s)

4. An orienteer runs 400 m directly east and then 500 m to the northeast (at a 45˚ angle from due east and from due north). Provide a graphic solution to show final displacement with respect to the starting position.

5. An orienteer runs north at 5 m/s for 120 s, and then west at 4 m/s for 180 s. Provide a graphic solution to show the orienteer’s resultant displacement.

6. Why are the horizontal and vertical components of projectile motion analyzed separately?

7. A soccer ball is kicked with an initial horizontal speed of 5 m/s and an initial vertical speed of 3 m/s. Assuming that projection and landing heights are the same and neglecting air resistance, identify the following quantities:

a. The ball’s horizontal speed 0.5 s into its flight

b. The ball’s horizontal speed midway through its flight

c. The ball’s horizontal speed immediately before contact with the ground

d. The ball’s vertical speed at the apex of the flight

e. The ball’s vertical speed midway through its flight

f. The ball’s vertical speed immediately before contact with the ground

8. If a baseball, a basketball, and a 71.2 N shot were dropped simultaneously from the top of the Empire State Building (and air resistance was not a factor), which would hit the ground first? Why?

9. A tennis ball leaves a racket during the execution of a perfectly horizontal ground stroke with a speed of 22 m/s. If the ball is in the air for 0.7 s, what horizontal distance does it travel? (Answer: 15.4m)

10. A trampolinist springs vertically upward with an initial speed of 9.2 m/s. How high above the trampoline will the trampolinist go? (Answer: 4.31 m)

1. Answer the following questions pertaining to the split times (in seconds) presented below for Ben Johnson and Carl Lewis during the 100 m sprint in the 1988 Olympic Games.

a. Plot velocity and acceleration curves for both sprinters. In what ways are the curves similar and different?

b. What general conclusions can you draw about performance in elite sprinters?

Johnson | Lewis | |
---|---|---|

10 m | 1.86 | 1.88 |

20 m | 2.87 | 2.96 |

30 m | 3.80 | 3.88 |

40 m | 4.66 | 4.77 |

50 m | 5.55 | 5.61 |

60 m | 6.38 | 6.45 |

70 m | 7.21 | 7.29 |

80 m | 8.11 | 8.12 |

90 m | 8.98 | 8.99 |

100 m | 9.83 | 9.86 |

2. Provide a trigonometric solution for Introductory Problem 4. (Answer: D = 832 m; ∠ = 25˚ north of due east)

3. Provide a trigonometric solution for Introductory Problem 5. (Answer: D = 937 m; ∠ = 50˚ west of due north)

4. A buoy marking the turn in the ocean swim leg of a triathlon becomes unanchored. If the current carries the buoy southward at 0.5 m/s, and the wind blows the buoy westward at 0.7 m/s, what is the resultant displacement of the buoy after 5 min? (Answer: 258m; ∠ = 54.5˚ west of due south)

5. A sailboat is being propelled westerly by the wind at a speed of 4 m/s. If the current is flowing at 2 m/s to the northeast, where will the boat be in 10 min with respect to its starting position? (Answer: D = 1.8 km; ∠ = 29˚ north of due west)

6. A Dallas Cowboy carrying the ball straight down the near sideline with a velocity of 8 m/s crosses the 50-yard line at the same time that the last Buffalo Bill who can possibly hope to catch him starts running from the 50-yard line at a point that is 13.7 m from the near sideline. What must the Bill’s velocity be if he is to catch the Cowboy just short of the goal line? (Answer: 8.35 m/s)

7. A soccer ball is kicked from the playing field at a 45˚ angle. If the ball is in the air for 3 s, what is the maximum height achieved? (Answer: 11.0 m)

8. A ball is kicked a horizontal distance of 45.8 m. If it reaches a maximum height of 24.2 m with a flight time of 4.4 s, was the ball kicked at a projection angle less than, greater than, or equal to 45˚? Provide a rationale for your answer based on the appropriate calculations. (Answer: >45˚)

9. A badminton shuttlecock is struck by a racket at a 35˚ angle, giving it an initial speed of 10 m/s. How high will it go? How far will it travel horizontally before being contacted by the opponent’s racket at the same height from which it was projected? (Answer: dv = 1.68 m; dh = 9.58 m)

10. An archery arrow is shot with a speed of 45 m/s at an angle of 10˚. How far horizontally can the arrow travel before hitting a target at the same height from which it was released? (Answer: 70.6 m)

• The metric system is the predominant standard of measurement in every major country in the world except the United States.

• Distance covered and displacement may be equal for a given movement. Or, distance may be greater than displacement, but the reverse is never true.

• The force of gravity produces a constant acceleration on bodies near the surface of the earth equal to approximately –9.81 m/s 2 .

• Neglecting air resistance, the horizontal speed of a projectile remains constant throughout the trajectory.

• The three mechanical factors that determine a projectile’s motion are projection angle, projection speed, and relative height of projection.

• A projectile’s flight time is increased by increasing the vertical component of projection velocity or by increasing the relative projection height.

Vertical Jump Height (cm) | Flight Time (s) |
---|---|

5 | 0.2 |

11 | 0.3 |

20 | 0.4 |

31 | 0.5 |

44 | 0.6 |

60 | 0.7 |

78 | 0.8 |

99 | 0.9 |

*Biomechanics of distance running,*Champaign, IL, 1990, Human Kinetics.

*The elite athlete,*New York, 1985, Spectrum Publications.

*Life span motor development*(3rd ed), Champaign, IL, 2001, Human Kinetics.

*Biomechanics of sport,*Boca Raton, FL, 1989, CRC Press.

*Biomechanics X-B,*Champaign, IL, 1987, Human Kinetics.

*Biomechanics of sport,*Boca Raton, FL, 1989, CRC Press.

*Biomechanics X-B,*Champaign, IL, 1987, Human Kinetics.

*Current research in sports biomechanics,*Basel, 1987, Karger.

Hay JG: Citius, altius, longius (faster, higher, longer): the biomechanics of jumping for distance, J Biomech 26:7, 1993.

McCoy RL: *Modern exterior ballistics:
the launch and flight dynamics of symmetric projectiles,* New
York, 1999, Schiffer.

*Provides a historical perspective on
early technological developments in the nineteenth century, including
the first ballistic firing tables.*

Saunders PU, Pyne DB, Telford RD, and Hawley JA: Factors affecting running economy in trained distance runners, Sports Med 34:465, 2004.

- www.physicsclassroom.com/Class/1DKin/1DKinTOC.html
*High-school-level tutorial on kinematics, including text and graphs.*

- www.physicsclassroom.com/Class/vectors/U3L2a.html
*High-school-level tutorial on projectile motion, including animations.*

- http://library.thinkquest.org/2779/
*Provides textual information and historical drawings of the first accurate description of projectile motion by Galileo, plus links to projectile animations, including a projectile water balloon game with sound effects.*

- www.cs.mtu.edu/~shene/COURSES/cs201/NOTES/chap02/projectile.html
*Provides a documented, downloadable computer program for calculating horizontal and vertical displacements, resultant velocity, and direction of a projectile.*

- http://galileo.phys.virginia.edu/classes/109N/more_stuff/Applets/ProjectileMotion/jarapplet.html
*Shows the trajectory of a projectile when the user enters the projection speed, angle, and height.*