Translational Motion
Linear Kinematics
Figure 8.1
Hamill & Knutzen (Ch 8)
Hay (Ch. 2 & 3)
Hay & Ried (Ch. 8 & 9)
Kreighbaum & Barthels (Module F)
or Hall (Ch. 2 & 10)
Quadrants in a Two-Dimensional
Reference System Figure 8.3
+y
Quadrant II (-,+)
-x
Quadrant I (+,+)
+x
Quadrant III (-,-)
Quadrant IV (+,-)
Position & Displacement
  Position defines an object’s location
in space.
  Displacement defines the change in
position that occurs over a given
period of time.
  Displacement is a vector
  Distance is a scalar
-y
Speed
Movements Occur Over Time
Knowledge of the
temporal patterns of
a movement is
critical in a kinematic
analysis since
changes in position
occur over time.
Speed is a scalar (m/s)
Speed = distance
Δtime
Δ = change in
1
Velocity
Velocity is a vector (m/s)
Velocity = Δposition(displacement)
Δtime
Velocity is designated by lowercase v
Time is designated by
lowercase t
Acceleration
Velocity
Fundamental units = LT-1
If we plot our displacement data on a
graph we are calculating the slope of
the line when we calculate velocity.
Units of Acceleration
Acceleration = Δvelocity
Δtime
 Units are m/s 2 or m.s-2
Acceleration is designated
by lowercase a
 Fundamental units => LT -2
It is used for both scalar
and vector quantities.
Acceleration
  If my velocity in the x-direction goes
from 3 m/s to 2 m/s in 0.05 seconds
what would my acceleration be?
  Answer: -20 m/s2
  Be careful of the term “deceleration”
Data Acquisition
Not covered in any
detail in most texts on
reserve.
2
Photographs
Where am I?
  If taken in the correct
plane photographs
can allow for later
evaluation of angles
and hence for a static
kinetic analysis.
  In dynamic situations
how do you know
you have the
extreme posture?
  The accurate
and complete
answer to this
question is not
as simple as it
may seem.
Video Systems
Opto-electronic systems
The location of the
joint centres of
rotation is entered
directly into the
computer.
Systems usually
come with
software that will
calculate velocity
and acceleration.
  There is a limited amount of quantitative
data that can be gleaned from a full-motion
video system.
  Stop frame capability does however allow
for a reasonably accurate assessment of
posture.
  60 frames/sec is more than adequate for
most movements but the real problem is
identifying joint centres of rotation and
calibration.
Q-Track and
Force Plate Data
Q-trac
markers
Previous Data Acquisition System in
Dr. Robinovitch’s Lab
3
Vectors and Scalars
Vector Components
  Scalars can be described by magnitude
  a = original vector
 E.g., mass, distance, speed, volume
  Vectors have both magnitude and
direction
a
θ
x-component
Displacement => Velocity
y-component
 E.g., velocity, force, acceleration
Displacement => Velocity
v = positionfinal - position initial
time at final displ. - time at initial displ.
B
v = xf - xi = xf - xi
tf - ti Δt
Δx = rise
A
y − component
a
x − component
cos θ =
a
sin θ =
  Vectors are represented by arrows
  See pages 305-308 for vector addition,
subtraction, and multiplication
Displ.
Figure 8.9
Δt = run
As (tf - ti ) is usually constant we
just use Δt.
Time
Finite Differentiation
Finite Differentiation
B (x2 , y2)
Δx
x1
x2
t1
t2
x3
x4
x5
t3
t4
t5
Δt
Displ.
A (x1, y1)
Time
4
Finite Differentiation
Finite Differentiation
x 2 − x1
Δt
y 2 − y1
Vy1.5 =
Δt
x 2 − x1
Δt
y 2 − y1
y 1.5 =
Δt
x1.5 =
Vx1.5 =
Sample Data
Finite Differentiation
Frame Time (s) Vert.Pos (y) Vel. (vy)
1
0.0000
0.00
2
0.0167
3
4
Frame Time (s) Vert.Pos (y) Vel. (vy)
8.98
1
0.0000
0.00
8.98
0.15
2
0.0167
0.15
4.19
0.0334
0.22
3
0.0334
0.22
2.99
0.0501
0.27
4
0.0501
0.27
(0.15 - 0.00) / 0.0167 = 8.98
(0.22 - 0.15) / 0.0167 = 4.19
Finite Differentiation
Finite Differentiation
x1
x2
Δt
V2-3
x1
x2
x3
x4
x5
x4
x5
V3
x3
2Δt
5
Finite Differentiation
Finite Differentiation
First central difference method
Frame Time (s) Vert.Pos (y) Vel. (vy)
x 3 − x1
x 2 = 2Δt
1
0.0000
0.00
0.00
2
0.0167
0.15
6.59
3
0.0334
0.22
3.59
4
0.0501
0.27
(0.22 - 0.00) / 0.0334 = 6.59
Acceleration
  Again if we are using coordinate systems
we use the following convention.
Horizontal ⋅ acceleration ⇒ x
Vertical ⋅ acceleration ⇒ y
Figure 8.18
Sample Problem
Time (s)
0.0
0.5
1.0
1.5
2.0
2.5
10.5
Displ. (m)
0.000
0.857
3.160
6.484
10.564
15.210
105.514
Figure 8.19
6
Time Displ. Time Displ. 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
0.000
0.857
3.160
6.484
10.56
15.21
20.26
25.60
31.130
36.77
42.480
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
48.21
53.95
59.68
65.42
71.16
76.89
82.63
88.37
94.11
99.84
105.51
Draw the
following
graphs (do not
use 1st central
difference)
Sampling Theory
d vs t
v vs t
a vs t
F vs v
Winter, 1979 (page 22-39)
Assume mass
of runner =
70 kg
How small should Δt be?
Instantaneous Velocity
Tangent
Displ.
This line would be a poor
estimate of the tangent for
this section of the curve.
Time
Δt
Sampling Theory
  Obviously the smaller Δt is the more
accurate you estimate of instantaneous
velocity.
  However, the smaller you try to get Δt the
more expensive it is going to be!
  Regular video at 60 frames/sec is good
for most applications.
7
Analogue to Digital
Synchronization (A to D)
  If you have force platform, video and EMG
data there can be a problem in
synchronizing the data.
  How do you know the time frames on each
data acquisition system match?
  No problem if all collected by computer, but
if some is collected on video and some on
the computer!
Aliasing Error
Fourier Transformation
Signal 1
Signal 2
f2
Filtering Raw Data
Sampling Theory
Differentiation and Noise
The differential of the line
between these markers is
much larger than the difference
in their location.
Red stars = true location of markers
Yellow stars = location of markers due to “noise”
8
Signal
vs Noise
Accelerometers
Integration
  Differentiating positional data to get
velocity and acceleration has been
covered.
  However, acceleration may be collected in
a biomechanical analysis.
  In this case, you may want to calculate
velocity and displacement data.
  This is the opposite of differentiation and
is known as integration.
Tri-Axial Accelerometers
  Accelerometers vary
considerably in resolution
and max. acceleration.
  Must be sure of planar
acceleration if using uniaxial accelerometers.
  Tri-axial accelerometers
are bulkier and much
more expensive.
  These can be rented
rather than purchased.
The Basic
Accelerometer:
A classical
second order
mass-spring
mechanical
system with
damping and
applied force
Vibration
Force Platform
Data
  Vibration is measured using accelerometers
and then various mathematical and statistical
techniques are used to quantify and interpret
the signal.
If you have force – and obviously F = ma, then
you can easily calculate the acceleration of the
body’s centre of gravity.
9
Finite Integration
  Finite differentiation methods are used with
digital data.
  Similarly, finite integration methods are used
with digital data.
  Finite differentiation calculates the slope of
the curve.
  Finite integration calculates the area under
the curve.
  Most often used with force-time curves – area
under the curve is mechanical impulse (more
in Linear Kinetics lecture)
If :→ a = Δv Δt
Then :→ Δv = aΔt
Acceleration
Finite Integration
€
Time
€
Hence area under curve = aΔt
Example
7
acc.
3
0
Time
6
8
  Area A equals
3m/s2 x 6s = 18 m/s
  Units! LT-2 x T = LT-1
  This is change in
velocity from 0-6 s.
  Area B is 14 m/s.
  Total change in
velocity from 0-8 s is
32 m/s.
Riemann Sum
  Finite differentiation
approximates the area
under curves as a
series of rectangles
  This is called the
Riemann sum (see
equation opposite)
  If Δt is small enough
this is an accurate
approximation
t30
∫ vxidt = ds
t1
30
ds =
∑ (v
xi
* dt)
i =1
Example above: Horizontal
velocity time curve with 30
time intervals. Integral
equals change in
displacement.
€
Integration is less sensitive to
errors due to “noise”
High frequency “noise” present
A
B
How fast can we go?
Kinematics of Running
Hamill & Knutzen, Chapter 8
(pages 319-323)
The slope of curve A varies greatly but the area
under the curve is not that different from curve B.
10
Stride Rate vs Stride Length
Running Kinematics
  Stride length (SL) and stride frequency (SF) are very
commonly studied kinematic parameters.
  Both SR & SL increase linearly (approx.) from a slow
jog up until 7 m/s.
  After this SR increases much more than SL.
  Support and non-support phases are also of interest.
  Support Phase: Jogging 68%, moderate sprint 54%,
full-sprint 47%
Mechanical Efficiency (Figure 8-27)
02 consumption
PSF = preferred
stride frequency
100 m vs 200 m
  The world record
for the 100m is?
  Women 10.49s (1988)
  Men 9.58s
  The world record
for the 200m is?
  Women 21.34s (1988)
  Men 19.19s
4 x 100 m Relay
Projectile Motion
  Best male 4 x 100m relay time = 37.10s
  This an average of 9.275s per 100m!
  Best female 4 x 100m relay time = 41.37s
  This an average of 10.34s per 100m!
  This is possible due to the fact an
acceleration phase is allowed within the
2nd, 3rd and 4th 100m segments.
11
Equations of Constant
Acceleration
1).......vf = vi + at
This is a re-arrangment of:
a = Δv
a = vf - vi
Δt
t
2).......d = vit + 0.5at2
3).......vf2 = vi2 + 2ad
Air Resistance
 
 
 
 
 
Can often be ignored but is often a
considerable factor.
Name a few examples where air (or fluid)
resistance in considerable.
Baseball, cycling, swimming, skydiving!
Name a few where it is negligible
Shot-put, long jump (possibly)?
Maximum Vertical Reach
Actual reach will be affected
by body anthropometry and
position.
In what ways?
Forces Influencing Projectiles
  Gravity
  Air resistance
  No other forces can
influence the flight
(trajectory) of the
projectile
  Air resistance is often
negligible
  Air resistance is
considered negligible
in this section of the
text
Without air
resistance
With air
resistance
Maximum Vertical
Displacement
  Relatively simplistic
  Height centre of gravity
(CG) reaches will be
determined by height of
CG at take-off and
vertical velocity of CG
at take-off.
Projecting for Horizontal
Distance
  It is a very common
performance objective
to project an object, or
the body, for
maximum horizontal
distance.
  Long jump, triple jump,
golf drive, football
punts.
12
Factors Affecting Trajectory
Optimum Angle of Projection
You need a large
horizontal velocity, but
if you sacrifice vertical
velocity you have little
time in the air.
Projection
Speed
Projection
Angle
Projection
Height
Optimum = 45o (without air resistance)
Range Equation
v
Range =
€
2
× sin θ × cosθ + v x
Vertical & Horizontal Components
are Independent
v
2
y
+ 2gh
g
  This can be arrived at via the use of the
equation, d = vit + 0.5at2 and knowing the
solution to a quadratic.
  There are many problems to work through in
the texts on reserve and the course workbook
Easier way to calculate range
Vertical
Time =
Horizontal
€
− vy ±
v
2
y
+ 2gh
g
Range = v x × time
Projecting for Accuracy
  Optimize velocity of
release, rather than
maximize, in sports like
darts, slow-pitch softball
  In other sports like
baseball, tennis, squash
and golf drives it is
desirable to have high
release velocities.
€
13
Optimum Angle of Projection
If the target is
above the release
point then the
optimum angle is
steeper (too low
and you don’t get
there!
If the target is
below the release
point then the
optimum angle is
shallower.
Horizontal Plane Targets
  Ideally a vertical
descent into the target
area is desirable.
  Again the horizontal
distance from the
target will determine
how closely one can
achieve this ideal.
Speed and Accuracy
Vertical Plane Targets
  Ideally would like
projectile projected to
target at 90o.
  However, the further
the projection distance,
for any given velocity,
the more arc (parabola)
needed on the
projectile.
Projecting the Body for
Accuracy
  Point targets in
space are often
used rather than a
physical target.
  Best examples are
in gymnastics and
other tumbling
activities.
Long Jumper’s Angle of Take-Off
  Many sports require
both accuracy & speed
(spike volleyball serve,
tennis serve, lacrosse
shot, etc.).
  One cannot maximize
projection speed in
some cases, but this
requirement cannot be
ignored.
14