# An Introduction to Linear

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14 Νοε 2013 (πριν από 4 χρόνια και 8 μήνες)

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B i o L a b
-

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
.

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

Williams & Wilkins. Pages 271
-
289.

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

Pages 47
-
62.