An Introduction to Linear

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Nov 14, 2013 (3 years and 7 months ago)

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B i o m e c h a n i c s T e a c h i n g & L e a r n i n g T o o l B o x

Linear Kinematics

An Introduction to Linear
Kinematics


Linear Kinematics


description of the motion of a body


the appearance of a motion with respect to time



Motion described in terms of (variables):


Distance, displacement, length (e.g. stride, stroke)


Time, cadence (e.g. stride frequency, stroke frequency)


Speed, velocity


Acceleration



Single point models


e.g. Centre of mass (CM) during running/jumping



Multi
-
segment models


e.g. Co
-
ordination of body segments during running/jumping


Kinematic Analysis

Distance & Displacement


Distance:


Length of path which a body covers during motion


Units: metre (m), centimeter (cm), kilometer (km)



Displacement:


The change in position of a body during motion


Units: metre (m), centimeter (cm), kilometer (km)



Distance is a
scalar
, and displacement is a
vector

variable

Speed and Velocity


Speed (scalar)


Length of path (distance)
divided by change in time
(

t
)




Average velocity (vector)


Change in position (

p
)
divided by change in time
(

t
)


Displacement (
d
) divided by
change in time (

t
)


Vector equivalent of linear
speed

If displacement = 50 m

Δt
d
=
Δt
Δp
=
v
If

t

= 5 s

v
= 50 / 5

=
10 m

s
-
1

Velocity


Units of velocity


m/s or m

s
-
1



Velocity is a vector


Magnitude and direction
calculated using Pythagoras
and trigonometry


The velocity of a swimmer in
a river is the vector sum of
the velocities of swimmer
and current.


Current
velocity

Swimmer’s
velocity

Resultant
velocity

Velocity


For human gait, speed
is the product of stride
length

and stride
velocity
.


Adults walk faster
using longer stride
lengths and faster
stride frequency.


Stride length in
children has great
variability.

Velocity


Runners traveling at a
slower pace tend to
increase velocity primarily
by stride ____?


At faster running speeds,
runners rely more on
increasing stride ____?


Most runners tend to
choose a combination of
stride length and stride
frequency that minimizes
physiological cost.

Best sprinters distinguished by high
stride ___ & short ground contact time.

Velocity


Pace
: rate of
movement, or
established rate of
locomotion.


Pace =

_time_



distance


Men’s world record
marathon pace =
4:37 min/mile
(2:03.38)


Women’s world
record marathon
pace = 5:30 min/mile

Position

(m)

Ben Johnson

Elapsed time

Johnson

Pace

Carl

Lewis

Interval time

Lewis

Pace

0

0

0

10

1.83 s

.183 s/m

1.89

.189 m/s

20

2.87 s

.104 s/m

2.96

.107 m/s

30

3.80 s

.093 s/m

3.90 s

.094 m/s

40

4.66 s

.086 s/m

4.79 s

.089 m/s

50

5.50 s

.084 s/m

5.65 s

.086 m/s

60

6.33 s

.083 s/m

6.48 s

.083 m/s

70

7.17 s

.084 s/m

7.33 s

.085 m/s

80

8.02 s

.085 s/m

8.18 s

.085 m/s

90

8.89 s

.087 s/m

9.04 s

.086 m/s

100

9.79 s

.090 s/m

9.92 s

.088 m/s

Men’s 100
-
m Dash 1988 Olympic Games

Velocity



Average velocity


Average velocity not
necessarily equal to
instantaneous velocity


Instantaneous velocity


Occurring at one instant in
time


Like an automobile
speedometer







Winner of the Men's 100 m at the
2004 Athens Olympics in 9.85 s


Average velocity = 100 / 9.85


=
10.15 m

s
-
1

2004 Olympic Men's 100 m

Kinematic analysis of 100 m sprint

Kinematic analysis of 100 m sprint

Velocity during 100 m

Average velocity 0
-
10 m

v

=
d

/

t

= 10 / 2.2 = 4.5 m

s
-
1


10
-
20 m

= 10 / 1.2 = 8.3 m∙s
-
1


20
-
30 m

= 10 / 0.8 = 12.5 m∙s
-
1


30
-
40 m

= 10 / 0.7 = 14.3 m∙s
-
1


40
-
50 m

= 10 / 0.8 = 12.5 m∙s
-
1

50
-
60 m

= 10 / 0.8 = 12.5 m∙s
-
1


60
-
70 m

= 10 / 0.7 = 14.3 m∙s
-
1


70
-
80 m

= 10 / 0.8 = 12.5 m∙s
-
1


80
-
90 m

= 10 / 0.9 = 11.1 m∙s
-
1


90
-
100 m

= 10 / 0.9 = 11.1 m∙s
-
1


Average Acceleration



Change in velocity (

v
) divided
by change in time (

t
)



Units


m/s/s or m/s
2

or m∙s
-
2



Vector


As with displacement & velocity,
acceleration can be resolved
into components using
trigonometry & Pythagorean
theorem



2 1
(v - v
v
a = =
t t
)

 
V
1

= 4.5 m
∙s
-
1

V
2

= 8.3 m
∙s
-
1

∆t

= 1.2
s

a

= (8.3
-

4.5) / 1.2 =
3.2 m
∙s
-
2

Acceleration during 100 m

Acceleration at start of race

a

= (
v
2
-

v
1
)
/

t

= (8.3
-

4.5) / 1.2





Positive Acceleration

=
3.2 m∙s
-
2

_____________________________________________________________________________________________________________________________
___
_

Acceleration during middle of race

a

= (
v
2
-

v
1
)

/

t

= (12.5
-

12.5) / 0.8





Constant Velocity

=

0

_____________________________________________________________________________________________________________________________
___
_

Acceleration at end of race

a

= (
v
2
-

v
1
)

/

t

= (11.1
-

14.3) / 0.9




Negative Acceleration

=
-
3.5 m∙s
-
2

Acceleration and Direction of
Motion


Complicating factor in understanding
acceleration is direction of motion of object.


When object moving in same direction
continually, accelerate often used to indicate
an increase in velocity and decelerate to
indicate a decrease in velocity.


If object changes direction, one direction is
positive, the opposite direction is negative.

Acceleration

Player running in negative direction increases negative
velocity results in negative acceleration.

Player begins to decrease velocity in negative direction has
positive acceleration.

Positive and negative accelerations can occur without
changing directions.

Motion in a negative direction

Increasing velocity

Decreasing velocity

Negative acceleration

Positive acceleration

Motion in a positive direction

Increasing velocity

Decreasing velocity

Negative acceleration

Positive acceleration

Summary


Variables used to describe motion are either:


Scalar (magnitude only: e.g. time, distance and speed)


Vector (magnitude and direction: e.g. displacement,
velocity and acceleration)



Displacement is the change in position of a body



Average velocity is the change in position divided by the
change in time



Average acceleration is the change in velocity divided by
the change in time



Enoka, R.M. (2002).
Neuromechanics of Human Movement

(3rd edition). Champaign, IL.: Human Kinetics. Pages 3
-
10
& 22
-
27.



Grimshaw, P., Lees, A., Fowler, N. & Burden, A. (2006).
Sport and Exercise Biomechanics
. New York: Taylor &
Francis. Pages 11
-
21.



Hamill, J. & Knutzen, K.M. (2003).
Biomechanical Basis of
Human Movement

(2nd edition). Philadelphia: Lippincott
Williams & Wilkins. Pages 271
-
289.



McGinnis, P.M. (2005).
Biomechanics of Sport and Exercise
(2nd edition). Champaign, IL.: Human Kinetics.

Pages 47
-
62.

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