Chapter 10: Linear Kinematics of Human Movement
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 change in time
=
d Δt
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 = change in time
Δv Δt
a = v2 - v 1 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 = v21 + 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 = v1 t V22 = v21
Equations of Constant Acceleration Vertical component: a = -9.81 m/s2
v2 = at D = ½ at2 V22 = 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
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 change in time
=
d Δt
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 = change in time
Δv Δt
a = v2 - v 1 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 = v21 + 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 = v1 t V22 = v21
Equations of Constant Acceleration Vertical component: a = -9.81 m/s2
v2 = at D = ½ at2 V22 = 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
