Chapter 10: Linear Kinematics of Human Movement

Objectives• Discuss the interrelationship among kinematicvariables• Correctly associate linear kinematic quantities withtheir units of measure• Identify & describe effects of factors governingprojectile trajectory• Explain why the horizontal and vertical componentsof projectile motion are analyzed separately• Distinguish between average & instantaneousquantities & identify circumstance which each is aquantity of interest

Linear Kinematic Quantities• Kinematics: describes appearance of motion• Kinetics: study of forces associated with motion• Linear kinematics: involves the study of theshape, form, pattern and sequencing of linearmovement through time• Qualitative: major joint actions & sequencing• Quantitative: Range of motion, forces, distanceetc.

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 & VelocitySpeed = length (or distance)change in timeVelocity (v) = change in position = Δ positionchange in timeΔ timev = displacement = dchange in timeΔ t

Speed & VelocityVelocity = position 2 - position 1time 2 - time 1• Velocity is a vector quantity– direction and magnitude of motion• Laws of vector algebra

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AccelerationAcceleration (a) = change in velocity =change in timeΔvΔta = v 2 - v 1ΔtWhen acceleration is zero, velocity is constant

Positive/Negative Acceleration

Average & InstantaneousQuantitiesInstantaneous :• Instantaneous valuesAverage:• Average velocity = final displacementtotal time

Velocity Curve for Sprinting

Velocity Curves for Two Sprinters

Kinematics of Projectile MotionBodies projected into the air are projectilesHorizontal & Vertical Components• Vertical is influenced by gravity• No force (neglecting air resistance) affectsthe horizontal• Horizontal relates to distance• Vertical relates to maximum height achieved

Kinematics of Projectile MotionInfluence of Gravity• Major influence of vertical component• Not the horizontal componentForce of Gravity:– Constant, unchanging– Negative acceleration (-9.81 m/s 2 )Apex:– The highest point in the trajectory

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Kinematics of Projectile MotionInfluence of Air Resistance• In a vacuum, horizontal speed of a projectileremain constant• Air resistance affects the horizontal speed ofa projectile• This chapter, velocity will be regarded asconstant

Factors InfluencingProjectile TrajectoryTrajectory:• Angle of projection• Projection speed• Relative height of projection

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Factors InfluencingProjectile TrajectoryAngle of Projection• General shapes– Perfectly vertical– Parabolic– Perfectly horizontal• Implications in sports• Air resistance may cause irregularities

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Factors InfluencingProjectile TrajectoryProjection speed:• Range:Relative Projection Height:

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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 MotionInitial velocity:• Horizontal component is constant– Horizontal acceleration = 0• Vertical component is constantly changing– Vertical acceleration = -9.81 m/s 2

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Equations ofConstant AccelerationGalileo’s Laws of constant accelerationv 2 = v 1 + atD = v 1 t + ½at 2V 2 2 = v 2 1 + 2 add = displacement; v = velocity;a = acceleration; t = timeSubscript 1 & 2 represent first or initial andsecond or final point in time

Equations ofConstant AccelerationHorizontal component : a = 0v 2 = v 1D = v 1 tV 2 2 = v2 1

Equations ofConstant AccelerationVertical component: a = -9.81 m/s 2v 2 = atD = ½ at 2V 2 2 = 2adVertical component at apex: v = 00 = v 2 1 + 2ad0 = 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 approximately42°– Long jump theoretical optimum isapproximately 43°; however, due to humanlimits, the actual angle for elite jumpers isapproximately 20° - 22°

Goals for Projectiles• Maximize total distance (golf)– Because the total distance (flight plus roll)is most important, trajectory angles arelower than 45°– Distance is controlled by the pitch of theclub• Driver ~ 10°

Goals for Projectiles• Optimize range and flight time (punt)– Maximum range occurs with 45° trajectory– Higher trajectory increases hang time withminimal sacrifice in distance– Lower trajectory usually results in longerpunt returns• Less time for kicking team to getdownfield to cover the punt returner

Goals for Projectiles• Maximize height (vertical jump)– Maximize height of COM at takeoff– Maximize vertical velocity by exertingmaximum 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 carryjumper over bar into pit– Typical takeoff velocity for elite highjumpers is approximately 45°

Goals for Projectiles• Minimize flight time (baseball throw)– Baseball players use low trajectories (closeto horizontal)– Outfielders often throw the ball on onebounce 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 orsequencing of linear motion with respect totime.• Linear kinematic quantities include the scalarquantities of distance and speed, and thevector quantities of displacement, velocity,and acceleration.• Vector quantities or scalar equivalent may beeither an instantaneous or an averagequantity

Summary• A projectile is a body in free fall that is affectedonly by gravity and air resistance.• Projectile motion is analyzed in terms of itshorizontal and vertical components.– Vertical is affected by gravity• Factors that determine the height & distance of aprojectile are: projection angle, projection speed,and relative projection height• The equation for constant acceleration can beused to quantitatively analyze projectile motion.

The End