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 the resultantdistance an object is removed
from its starting point.
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 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.
Examples of displacement–time graphs for uniform
and nonuniform motion. The velocity in (a), (b), and (c) is constant.
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
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
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
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
An acceleration curve showing a constant acceleration
(uniform change in velocity) in (a), zero acceleration (no change
in velocity) in (b), varying acceleration in (c), and constant deceleration (uniform
slowing) in (d).
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,
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
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.
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
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:
Table 11.1 Effect of Gravity
on a Freely Falling Object ||Download (.pdf)
Table 11.1 Effect of Gravity
on a Freely Falling Object
|U.S.||1 sec||16 ft||32 ft/sec||16 ft/sec|
|a = 32 ft/sec2||2 sec||64 ft||64 ft/sec ||32 ft/sec|
|3 sec||144 ft||96 ft/sec||48 ft/sec|
|4 sec||256 ft||128 ft/sec||64 ft/sec|
|5 sec||400 ft ||160 ft/sec||80 ft/sec|
|Metric||1 sec||4.9 m||9.8 m/sec||4.9 m/sec|
|a = 9.81 m/sec2||2 sec||19.6 m||19.6 m/sec||9.8 m/sec|
|3 sec||44.1 m||29.4 m/sec||14.7 m/sec|
|4 sec||78.4 m||39.2 m/sec||19.6 m/sec|
|5 sec||122.5 m||49 m/sec||24.5 m/sec|
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?
Downward Landing Velocity
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
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.
Objects projected upward are decelerated by the downward
force of gravity at the same rate as those allowed to fall downward
are accelerated. Both objects will cover the same distance in the same
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).
Effect of gravity on the flight of a projectile.
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).
A horizontally projected object and a free-falling object
released from the same height will land at the same time but in
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.
Magnitude of horizontal and vertical velocities during
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).
The angle of projection influences the horizontal and
vertical distance covered by a projectile. Complementary angles
of projection produce the same horizontal displacement.
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 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).
Table 11.2 Relationships
between Rotary and Linear Motion
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.
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.
Variations in the angular displacement of a golf club
over equal time intervals during the execution of a golf drive.
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.
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.
Lever PA > PB > PC. Although the angular displacement
for all three levers is the same, the linear displacement at the end
of the longer levers is greater than that at the end of the shorter
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.
Increasing the length of the radius decreases the angular
velocity when the linear velocity remains constant.
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
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
Long levers and high angular velocities result in high
linear velocities at the ends of the levers. Thus the tennis racket,
an extension of a long body lever, is able to impart high linear
velocity to the ball.