Chapter 10: Linear Kinematics of Human Movement

Basic Biomechanics, 4th 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 = position2 - position1

time2 - time1

•  Velocity is a vector quantity

– direction and magnitude of motion

•  Laws of vector algebra

10-2

Acceleration

Acceleration (a) = change in velocity = Δv

change in time Δt

a = v2 - v1

Δt

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/s2)

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 = 450

– ↑ 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/s2

10-17

Equations of Constant Acceleration

Galileo’s Laws of constant acceleration

v2 = v1 + at

D = v1t + ½at2

V22 = v2

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

v2 = v1

D = v1t

V22 = v2

1

Equations of Constant Acceleration

Vertical component: a = -9.81 m/s2 v2 = at D = ½ at2 V2

2 = 2ad Vertical component at apex: v = 0

0 = v21 + 2ad

0 = v1 + 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