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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
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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
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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.
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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
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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
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Speed & Velocity
Velocity = position2 - position1
time2 - time1
• Velocity is a vector quantity
– direction and magnitude of motion
• Laws of vector algebra
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10-2
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Acceleration
Acceleration (a) = change in velocity = Δv
change in time Δt
a = v2 - v1
Δt
When acceleration is zero, velocity is constant
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Positive/Negative Acceleration
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Average & Instantaneous Quantities
Instantaneous :
• Instantaneous values
Average:
• Average velocity = final displacement
total time
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Velocity Curve for Sprinting
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Velocity Curves for Two Sprinters
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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
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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
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10-6
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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
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Factors Influencing Projectile Trajectory
Trajectory:
• Angle of projection
• Projection speed
• Relative height of projection
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10-9
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Factors Influencing Projectile Trajectory
Angle of Projection
• General shapes
– Perfectly vertical
– Parabolic
– Perfectly horizontal
• Implications in sports
• Air resistance may cause irregularities
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10-10
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Factors Influencing Projectile Trajectory
Projection speed:
• Range:
Relative Projection Height:
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10-14
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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
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Range at Various Angles
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Analyzing Projectile Motion
Initial velocity:
• Horizontal component is constant
– Horizontal acceleration = 0
• Vertical component is constantly changing
– Vertical acceleration = -9.81 m/s2
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10-17
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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
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Equations of Constant Acceleration
Horizontal component : a = 0
v2 = v1
D = v1t
V22 = v2
1
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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
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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)
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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°
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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°
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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
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Goals for Projectiles
• Maximize height (vertical jump)
– Maximize height of COM at takeoff
– Maximize vertical velocity by exerting maximum vertical force against ground.
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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°
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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
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Goals for Projectiles
• Accuracy (basketball shot)
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Projecting for Accuracy
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Minimum Speed Trajectory
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Angle of Entry
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Margin for Error
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Free Throw Optimum Angle
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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
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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.
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The End