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