Chapter 10:
Linear Kinematics of
Human Movement
Basic Biomechanics, 4
th
edition
Susan J. Hall
Presentation Created by
TK Koesterer, Ph.D., ATC
Humboldt State University
Objectives
•
Discuss the interrelationship among kinematic
variables
•
Correctly associate linear kinematic quantities with
their units of measure
•
Identify & describe effects of factors governing
projectile trajectory
•
Explain why the horizontal and vertical components
of projectile motion are analyzed separately
•
Distinguish between average & instantaneous
quantities & identify circumstance which each is a
quantity of interest
Linear Kinematic Quantities
•
Kinematics:
describes appearance of motion
•
Kinetics:
study of forces associated with motion
•
Linear kinematics:
involves the study of the
shape, form, pattern and sequencing of linear
movement through time
•
Qualitative:
major joint actions & sequencing
•
Quantitative:
Range of motion, forces, distance
etc.
Distance & Displacement
•
Measured in units of length
–
Metric: meter, kilometer, centimeter, etc.
–
English: inch, foot, yard & mile
•
Distance:
–
Scalar quantity
•
Linear displacement:
–
Vector quantity: length & direction
(compass directions, left, right, up, & down,
or positive & negative
Speed & Velocity
Speed = length (or distance)
change in time
Velocity (v) = change in position =
Δ
position
change in time
Δ
time
v = displacement = d
change in time
Δ
t
Speed & Velocity
Velocity = position
2
 position
1
time
2
 time
1
•
Velocity is a vector quantity
–
direction and magnitude of motion
•
Laws of vector algebra
102
Acceleration
Acceleration (a) = change in velocity =
Δ
v
change in time
Δ
t
a = v
2
 v
1
Δ
t
When acceleration is zero, velocity is constant
Positive/Negative Acceleration
Average & Instantaneous
Quantities
Instantaneous :
•
Instantaneous values
Average:
•
Average velocity = final displacement
total time
Velocity Curve for Sprinting
Velocity Curves for Two Sprinters
Kinematics of Projectile Motion
Bodies projected into the air are
projectiles
Horizontal & Vertical Components
•
Vertical is influenced by gravity
•
No force (neglecting air resistance) affects
the horizontal
•
Horizontal relates to distance
•
Vertical relates to maximum height achieved
Kinematics of Projectile Motion
Influence of Gravity
•
Major influence of vertical component
•
Not the horizontal component
Force of Gravity:
–
Constant, unchanging
–
Negative acceleration (9.81 m/s
2
)
Apex:
–
The highest point in the trajectory
106
Kinematics of Projectile Motion
Influence of Air Resistance
•
In a vacuum, horizontal speed of a projectile
remain constant
•
Air resistance affects the horizontal speed of
a projectile
•
This chapter, velocity will be regarded as
constant
Factors Influencing
Projectile Trajectory
Trajectory:
•
Angle of projection
•
Projection speed
•
Relative height of projection
109
Factors Influencing
Projectile Trajectory
Angle of Projection
•
General shapes
–
Perfectly vertical
–
Parabolic
–
Perfectly horizontal
•
Implications in sports
•
Air resistance may cause irregularities
1010
Factors Influencing
Projectile Trajectory
Projection speed:
•
Range:
Relative Projection Height:
1014
Optimum Projection Conditions
•
Maximize the speed of projection
•
Maximize release height
•
Optimum angle of projection
–
Release height = 0, then angle = 45
0
–
↑
Release height, then
↓
angle
–
↓
Release height, then
↑
angle
Range at Various Angles
Analyzing Projectile Motion
Initial velocity
:
•
Horizontal component is constant
–
Horizontal acceleration = 0
•
Vertical component is constantly changing
–
Vertical acceleration = 9.81 m/s
2
1017
Equations of
Constant Acceleration
Galileo’s Laws of constant acceleration
v
2
= v
1
+ at
D = v
1
t + ½at
2
V
2
2
= v
2
1
+ 2 ad
d = displacement; v = velocity;
a = acceleration; t = time
Subscript 1 & 2 represent first or initial and
second or final point in time
Equations of
Constant Acceleration
Horizontal component : a = 0
v
2
= v
1
D = v
1
t
V
2
2
= v
2
1
Equations of
Constant Acceleration
Vertical component: a = 9.81 m/s
2
v
2
= at
D = ½ at
2
V
2
2
= 2ad
Vertical component at apex: v = 0
0 = v
2
1
+ 2ad
0 = v
1
+ at
Goals for Projectiles
•
Maximize range (shot put, long jump)
•
Maximize total distance (golf)
•
Optimize range and flight time (punt)
•
Maximize height (vertical jump)
•
Optimize height and range (high jump)
•
Minimize flight time (baseball throw)
•
Accuracy (basketball shot)
Goals for Projectiles
•
Maximize range (shot put, long jump)
–
Shot put optimum angle is approximately
42
°
–
Long jump theoretical optimum is
approximately 43°; however, due to human
limits, the actual angle for elite jumpers is
approximately 20°  22°
Goals for Projectiles
•
Maximize total distance (golf)
–
Because the total distance (flight plus roll)
is most important, trajectory angles are
lower than 45
°
–
Distance is controlled by the pitch of the
club
•
Driver ~ 10
°
Goals for Projectiles
•
Optimize range and flight time (punt)
–
Maximum range occurs with 45
° trajectory
–
Higher trajectory increases hang time with
minimal sacrifice in distance
–
Lower trajectory usually results in longer
punt returns
•
Less time for kicking team to get
downfield to cover the punt returner
Goals for Projectiles
•
Maximize height (vertical jump)
–
Maximize height of COM at takeoff
–
Maximize vertical velocity by exerting
maximum vertical force against ground.
Goals for Projectiles
•
Optimize height and range (high jump)
–
Basic goal is to clear maximum height
–
Horizontal velocity is necessary to carry
jumper over bar into pit
–
Typical takeoff velocity for elite high
jumpers is approximately 45
°
Goals for Projectiles
•
Minimize flight time (baseball throw)
–
Baseball players use low trajectories (close
to horizontal)
–
Outfielders often throw the ball on one
bounce with minimal loss of velocity
Goals for Projectiles
•
Accuracy (basketball shot)
Projecting for Accuracy
Minimum Speed Trajectory
Angle of Entry
Margin for Error
Free Throw Optimum Angle
Summary
•
Linear kinematics is the study of the form or
sequencing of linear motion with respect to
time.
•
Linear kinematic quantities include the scalar
quantities of distance and speed, and the
vector quantities of displacement, velocity,
and acceleration.
•
Vector quantities or scalar equivalent may be
either an instantaneous or an average
quantity
Summary
•
A projectile is a body in free fall that is affected
only by gravity and air resistance.
•
Projectile motion is analyzed in terms of its
horizontal and vertical components.
–
Vertical is affected by gravity
•
Factors that determine the height & distance of a
projectile are: projection angle, projection speed,
and relative projection height
•
The equation for constant acceleration can be
used to quantitatively analyze projectile motion.
The End
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