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

10-2

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

10-6

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

10-9

Factors Influencing

Projectile Trajectory

Angle of Projection

•

General shapes

–

Perfectly vertical

–

Parabolic

–

Perfectly horizontal

•

Implications in sports

•

Air resistance may cause irregularities

10-10

Factors Influencing

Projectile Trajectory

Projection speed:

•

Range:

Relative Projection Height:

10-14

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

10-17

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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