Chapter 12
Linear Kinetics of Human
Movement
Basic Biomechanics, 6th edition
Susan J. Hall
Presentation Created by
TK Koesterer, Ph.D., ATC
Humboldt State University
Objectives
• Identify Newton’s laws of motion and gravitation and
describe practical illustrations of the laws
• Explain what factors affect friction and discuss the role
of friction in daily activities and sports
• Define impulse and momentum and explain the
relationship between them
• Explain what factors govern the outcome of a collision
between two bodies
• Discuss the interrelationship among mechanical work,
power, and energy
• Solve quantitative problems related to kinetic concepts
Newton’s Laws
• Isaac Newton
developed the
theories of
gravitation in 1666,
when he was 23
years old.
• In 1686 he
presented his three
laws of motion.
Newton’s Laws
Law of Inertia
• A body will maintain a
state of rest or constant
velocity unless acted on
by an external force that
changes its state.
• Inertia is the property of
a body that causes it to
remain at rest if it is at
rest or continue moving
with a constant velocity
unless a force acts on it.
Newton’s First Law of Motion
• Law of Inertia
Every body will remain in a state of
rest or constant motion (velocity) in a
straight line unless acted on by an
external force that changes that state
• A body cannot be made to change
its speed or direction unless acted
upon by a force(s)
• Difficult to prove on earth due to the
presence of friction and air
resistance
Newton’s Laws
Law of Acceleration
• A force (F) applied to a body
causes an acceleration (a) of
that body of a magnitude
proportional to the force, in the
direction of the force, and
inversely proportional to the
body’s mass (m).
• Forms the link between force
and motion:
• Force = mass x acceleration or
acceleration =
Force
mass
Applications of Newton’s 2nd Law
• Assuming mass remains
constant, the greater the
force the greater the
acceleration
F = 500 N
a=?
m = 1.5 kg
• Acceleration is inversely
proportional to mass
– if force remains the same
and mass is halved, then
acceleration is doubled
– if force remains the same
and mass is doubled, then
acceleration is halved
F=m×a
F
500
a=
=

m
1.5
a = 333 m·s-2
Newton’s Laws
Law of Reaction
• For every action, there
is an equal and
opposite reaction
• When one body exerts
a force on a second, the
second body exerts a
reaction force that is
equal in magnitude and
opposite in direction of
the first body
Newton’s Laws
Law of Reaction
• During gait, every
contact of a foot with
the ground generates
an upward reaction
force by the ground
called (GRF).
• GRF has both
horizontal and vertical
components.
• What is magnitude of
vertical GRF running?
Newton’s Laws
Law of Gravitation
• All bodies are attracted to one another
with a force proportional to the product
of the masses and inversely
proportional to the square of the
distance between them
• Fg = G(m1m2 / d2)
–
–
–
–
–
Fg = Force of attraction
Earth mass = 5.972 x 1024 kg
Earth radius = 6.378 x 103 km
Greater mass, greater attraction
Greater distance, less attraction
Implications of Newton’s
Law of Gravitation
• Mass
– Greater mass =
greater gravitational force
– Smaller mass =
lower gravitational force
• Distance
– Greater distance =
smaller gravitational force
– Smaller distance =
greater gravitational force
• Most bodies in sport have
relatively small mass
– Attractive force between
them can be considered
negligible
Newton’s Laws
Law of Gravitation
Acceleration Because of Gravity at Sea
Level by Latitude
Latitude Acceleration
due to Gravity
Sample
Locations
0
9.780
Ecuador,
Kenya
15
9.784
Philippines,
Guatemala
30
9.793
Texas, Israel
45
9.806
Oregon,
France
60
9.819
Alaska,
Sweden
75
9.829
Greenland
90
9.832
North Pole
Gravitational variations largely accounted
for by the equatorial bulge of the earth
rather than altitude above sea level.
Weight
• Weight (W) is the attractive force
between the earth and any body
in contact with it or close to its
surface
• Product of the mass (m) of the
body and the acceleration
caused by the attractive force
between it and the earth
(g = 9.81 m·s-2)
rpoles
requator
i.e. W = m × g
• Gravity is based on:
– Mass of bodies
– Distance between bodies
r=radius of earth
requator > rpoles
Gequator < Gpoles
Wequator < Wpoles
Mechanical Behavior of Bodies in Contact
Friction
• Friction is a force that acts
parallel to the surfaces in
contact and opposite to
the direction of motion.
• If there is no motion,
friction acts in opposite
direction to any force that
“tends to produce motion”.
• Friction is a necessity for
and a hindrance to motion.
Magnitude of
friction forces
determine
relative ease
or difficulty of
motion for
two objects in
contact.
12-5
Mechanical Behavior of Bodies in Contact
Friction
• Starting friction is greater than moving friction.
• It takes more force to start moving an object
than it does to keep it moving.
Maximum static friction (Fm):
• As magnitude of applied force becomes
greater and greater, magnitude of opposing
friction force increases to a critical point.
Kinetic (sliding) friction (Fk):
• Magnitude of kinetic friction remains constant.
Mechanical Behavior of Bodies in Contact
Friction
Static
Fm = sR
Dynamic
Fk = kR
Friction
For static bodies,
friction is equal to
the applied force.
For bodies in
motion, friction is
constant and less
than maximum
static friction.
Applied external force
Mechanical Behavior of Bodies in Contact
Friction
Ff = R
Ff = frictional force;  = coefficient of friction; R = normal
(┴) reaction force
Coefficient of friction: number that serves as index
• Coefficient of static friction (s):
• Coefficient of kinetic (sliding) friction (k) :
Normal reaction force (┴): force acting . . .
Rolling friction: influence by weight, radius,
deformability of rolling object, plus 
Mechanical Behavior of Bodies in Contact
Friction
Ff = R
Amount of friction changed by altering μ.
– Gloves in racquetball, golf, batting
– Wax on surfboard or cross country skis
– Rosin on dance floor
• Provides large  s
• Provides significantly smaller k
Amount of friction changed by altering R.
– Press surfaces together (+ or – weight)
Mechanical Behavior of Bodies in Contact
Friction
Material
Hardwood on hardwood
Steel on steel
Steel on steel (oiled)
Rubber on dry concrete
Rubber on wet concrete
Starting Friction Sliding Friction
0.40
0.58
0.13
2.0
1.5
0.25
0.20
0.13
1.0
0.97
Synovial fluid present in many joints reduces friction
between articulating bones.
Mechanical Behavior of Bodies in Contact
Friction
• Artificial turf and cleats
coefficient of friction
cause more injuries?
Shoe traction dry
• μ = .90 to 1.50
Shoe traction wet
• μ = 1.10 to 1.50
FIFA recommendation
• μ = 0.35 to 0.75
Mechanical Behavior of
Bodies in Contact
Momentum is quantity of motion a body possesses.
Linear Momentum: M or p
• M=m•v
• Units: kg • m/s (or slug • ft/s)
• Downhill skier example: 55 kg • 30 m/s = 1650 kg • m/s
Newton’s laws can be expressed in terms of momentum.
Mechanical Behavior of Bodies
in Contact
• Newton’s first law (inertia) states that in the
absence of external forces the momentum of
an object remains constant.
M = constant
Principle of Conservation of Momentum
• Newton’s second law (acceleration) states
that the rate of change of momentum equals
the net external force acting on it.
Mechanical Behavior of Bodies in
Contact
Newton’s third law (action-reaction) may be stated
in momentum terms as whenever two bodies
exert forces on one another, the resulting
changes of momentum are equal and opposite.
Principle of conservation of momentum:
In the absence of external forces, the total
momentum of a given system remains constant.
M1 = M2
Mechanical Behavior of
Bodies in Contact
Principle of conservation of
momentum:
In the absence of external
forces, the total
momentum of a given
system remains constant.
M1 = M2 or (mv)1 = (mv)2
• Initial momentum (M1) of
objects before collision
• Final momentum (M2) of
objects after collision
Mechanical Behavior of Bodies in
Contact
Golf Ball
and Club
example:
Golf Ball mass = .045 kg, Golf Club mass = .200 kg
GBv1 = 0, GCv1= 270 m/s; GBv2=360 m/s, GCv2= ?
• How much does Golf Club slow down?
Momentum before
Momentum after impact
.045 · 0 + .2 · 270 m/s = .045 · 360 m/s + .2 · GCv2
0 + 54 kg · m/s =
16.2 kg · m/s + .2 kg· GCv2
37.8 kg · m/s  .2 kg = 189 m/s = GCv2
Golf Club Velocity Decreases (189 – 270) = - 81 m/s
Mechanical Behavior of Bodies in Contact
Impulse
Changes in momentum depend on force and
length of time during which force acts.
Impulse: product of force and time interval the force acts
• Impulse (J) = F  t
Derived from Newton’s Second law:
• F = ma; (a = v2 - v1 / t)
• F = m ([v2 - v1] / t) = (mv2 – mv1)/ t
• Ft = (mv2) - (mv1)
• Ft = M2 – M1 = M
This is the impulse-momentum relationship.
Mechanical Behavior of Bodies in Contact
Impulse
• Bunch start results in
clearing blocks sooner
but with less velocity.
• Highest proportion of
best runs are from 16-in
stance.
• Elongated stance of 26in results in greater
velocity leaving blocks
but lost within 10 yds
Block
Spacing
11 in.
16 in.
21 in.
26 in.
Time on
Blocks, i.e.
time from gun
to foot leaving
0.345 s
0.374 s
0.397 s
0.426 s
Block
Velocity, i.e.
horizontal
velocity as
leave blocks
6.63 ft/s
7.41 ft/s
7.50 ft/s
7.62 ft/s
Time to 10 yd
(9.1 m)
2.070 s
2.054 s
2.041 s
2.049 s
Time to 50 yd
(45.7 m)
6.561 s
6.479 s
6.497 s
6.540 s
Adapted from Henry, F. M. (1952) Research
Quarterly, 23:306.
Impulse
Since Impulse = F x t, i.e.
the amount of force applied
during a period of time,
impulse is the area under
the force curve.
Which jump generated
greater change in
momentum (vertical
velocity)?
12-10
Impulse
Horizontal (sagittal) Plane
• If initial negative impulse <
push off positive impulse,
horizontal velocity
increased.
• If initial negative & push off
impulse equal, no change
in horizontal velocity.
• If initial negative impulse >
push off positive impulse,
horizontal velocity
decreased.
Mechanical Behavior of Bodies in Contact
Impact
Impact: collision of two bodies over small time
Elasticity: an object’s ability to return to its
original size and shape when outside forces
are removed.
Perfectly elastic impact: relative velocities same
Perfectly plastic impact: at least one body loses
velocity, bodies don’t separate
Most impacts are neither perfectly elastic nor
perfectly plastic.
Mechanical Behavior of Bodies
in Contact
The differences in two Ball velocities before impact
u1
u2
balls’ velocities before
impact is proportional
to the difference in
their velocities after
impact. The factor of
v1
v2
proportionality is the
Ball velocities after impact
coefficient of
v1 - v2 = -e ( u1 - u2)
restitution.
Mechanical Behavior of Bodies in Contact
Impact
Impact (cont.)
Coefficient of restitution:
When two bodies undergo a direct collision,
the difference in their velocities immediately
after impact is proportional to the difference in
their velocities immediately before impact
-e = relative velocity after impact = v1 - v2
relative velocity before impact
u 1 - u2
Elasticity
• In case of impact
between moving body
and stationary one,
e= hb/hd
• Coefficient of
restitution describes
interaction between
two bodies, not a
single object or
surface.
Elasticity
Factors Affecting C of R
1. Velocities
2. Temperature
3. Material
4. Spin
C of R
Concrete Wood
Basketball
0.57
0.80
Golf ball
0.89
0.66
Racquetball
0.86
0.84
Baseball
0.57
0.45
Elasticity - Spin
• Magnitude of horizontal
forces exerted on ball
are influenced by
amount of spin.
• Horizontal velocity of
points on ball sum of 2
component velocities –
translational component
& rotational component.
Elasticity - Spin
• The angle of
incidence
(approach) equals
the angle of
reflection in perfectly
elastic impact with
no spin imparted.
Elasticity - Spin
• When topspin applied,
translational component
of part of ball that
contacts the floor is
offset by rotational
component; evoked
frictional force is less as
is the decrease in ball’s
forward velocity.
Elasticity - Spin
• When backspin applied,
the translational &
rotational components
complement each
other, the evoked
frictional reaction is
increased, and postimpact velocity is less.
• Backspin causes ball to
bounce more slowly &
at higher angle.
Summary
• Linear kinetics is the study of the forces associated
with linear motion
• Friction is a force generated at the interface of two
surfaces in contact
• Magnitudes of maximum static friction and kinetic
friction are determined by the coefficient of friction
and normal reaction force pressing the two surfaces
together.
• Linear momentum is the product of an object’s mass
and its velocity
Summary
• Total momentum in a given system remains
constant barring the action of external forces
• Changes in momentum result from impulses,
external forces acting over a time interval
• The elasticity of an impact governs the
amount of velocity in the system following
impact
• The relative elasticity of is represented by the
coefficient of restitution