19_KINE426_Linear_Kinematics2

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1 Andrew Nordin, PhD Assistant Professor, Division of Kinesiology KINE 426 Exercise Biomechanics Linear Kinematics Chapter 8 Kinematics • Kinematic data • Time series • Average vs. instantaneous • Position, velocity, acceleration • Graphing • Calculus relationships

2 Kinematics Kinematics Linear Angular Scalar Vector Distance Speed Acceleration Displacement Velocity !"##$ = !"#$%&'() !"#$% !"#$% !"# = !"#$%&'()*)+, !"#$% or ! ! ! ! ! ! ! − ! ! !""#$#%!&'( ! = !"#$%&'() !"#$% ! ! − ! ! ! ! − ! ! • Spatial & temporal components of motion • Ignores forces causing motion Magnitude Magnitude & direction or Projectile Motion • Discrete point analysis

3 Projectile Motion • Uniform acceleration equations – 3 equations allow kinematic variables to be determined • Position • Velocity • Acceleration • Time !"#$% !"# = !"#$%&'()*)+, !"#$% or ! ! ! ! ! ! ! − ! ! Projectile Motion • Uniform acceleration equations – 3 equations allow kinematic variables to be determined • Position • Velocity • Acceleration • Time ࠵? ! " = ࠵? $ " + 2࠵?࠵? ࠵? ࠵? = ࠵? ࠵?’ ࠵?࠵? ࠵? = ࠵? $ ࠵? + ( " ࠵?࠵? " a g = -9.81m/s 2

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4 Projectile Motion R ∆ d y_up t up t down t flight = t up + t down + projection height v yf_up = 0 = v yi_down ∆ d y_down d yi_up d yf_down = 0 v yi_up v yf_down v x • Uniform acceleration equations – 3 equations allow kinematic variables to be determined • Position • Velocity • Acceleration • Time ࠵? ! " = ࠵? $ " + 2࠵?࠵? ࠵? ࠵? = ࠵? ࠵?’ ࠵?࠵? ࠵? = ࠵? $ ࠵? + ( " ࠵?࠵? " a g = -9.81m/s 2 Most human movement a ≠ 0 Kinematic Data • Frames • Digitization – Scaling factor: Pixels per Real World Measurement • Sampling rate (frame rate) – fs = 1/Δ t → Hz = 1/s 1 2 3 4 5 Δ t Δ t Δ t Δ t

5 Kinematic Data 1 2 3 4 5 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 Hip height (m) Frame Frame Hip Height (m) 1 0.89 2 0.82 3 0.55 4 0.82 5 1.24 Time Series • Series (list) of numbers • Temporally arranged • Time between cells determined by sampling rate 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 0.00 0.25 0.50 0.75 1.00 Hip height (m) Time (s) fs = 1/Δ t → Hz = 1/s fs = 1/(0.005s) = 200Hz

6 Position Velocity !"#$% !"# = !"#$%&'()*)+, !"#$% or ! ! ! ! ! ! ! − ! ! d 1 d 2 t 1 t 2 Acceleration (m) (m/s) (m/s 2 ) v = (1.14m – 1.13m) = 0.01m = 2 m/s (1.305s – 1.300s) 0.005s !""#$#%!&'( ! = !"#$%&'() !"#$% ! ! − ! ! ! ! − ! ! or Δ position with respect to t Δ velocity with respect to t Position, velocity, acceleration 1 2 3 4 5 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 Hip height (m) Frame t 1 t 2 d 1 d 2 Average velocity Slope Rise Run d 2 d 1 t 1 t 2 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 0.00 0.25 0.50 0.75 1.00 Hip height (m) Time (s) Secant Intersects line @ 2 points Tangent Intersects line @ single point Instantaneous velocity Average vs. Instantaneous

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7 • Average vs. Instantaneous – Average Δ in position with respect to t between frames – Instantaneous rate of change @ a given frame • First central difference method – Average Δ with respect to t – Aligned with actual frame d 1 d 2 d 3 d 4 d 5 d 6 | ! Δ t " | ! Δ t " | ! Δ t " | ! Δ t " | ! Δ t " | t 1 t 2 t 3 t 4 t 5 t 6 v 1 " 2 v 4 t 1 t 2 d 1 d 2 v 1→2 d 1 d 3 v 2 First central difference method ? No frame • Information loss – Position → Velocity → Acceleration – Collect additional frames • Before & after movement of interest d 1 d 2 d 3 d 4 d 5 d 6 | ! Δ t " | ! Δ t " | ! Δ t " | ! Δ t " | ! Δ t " | t 1 t 2 t 3 t 4 t 5 t 6 v 1 " 2 v 4 ! ! = ! ! ! ! − ! ! ! ! ! ! ! ! − ! ! ! ! ! ! = ! ! ! ! − ! ! ! ! ! ! ! ! − ! ! ! ! ! ! = ! ! − ! ! ! ! − ! ! First central difference method

8 • Sampling rate – Frames/s – Hz • Temporal resolution – 1/Frames/s – Time between frames 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 0.00 0.25 0.50 0.75 1.00 Hip height (m) Time (s) 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 0.00 0.25 0.50 0.75 1.00 Hip height (m) Time (s) Average vs. Instantaneous Sampling rate ↑ Time between frames ↓ Acceleration Position Velocity !"#$% !"# = !"#$%&'()*)+, !"#$% or ! ! ! ! ! ! ! − ! ! !""#$#%!&'( ! = !"#$%&'() !"#$% ! ! − ! ! ! ! − ! ! or ! ! = ! ! ! ! − ! ! ! ! ! ! ! ! − ! ! ! ! ! ! = ! ! ! ! − ! ! ! ! ! ! ! ! − ! ! ! ! !"#$% = !"#$ !"# 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 0.00 0.25 0.50 0.75 1.00 Hip height (m) Time (s) tangent Position, velocity, acceleration

9 Acceleration Position Velocity 0.00 0.50 1.00 1.50 Hip height (m) -1.00 0.00 1.00 2.00 3.00 Hip velocity (m/s) -15.00 -10.00 -5.00 0.00 5.00 10.00 15.00 Hip acceleration (m/s 2 ) Time (s) Position, velocity, acceleration Derivative relationships • Calculus – Derivative / Differentiation – With respect to time – Slope of a line (tangent or secant) ! ! = !" !" = ∆ ! ∆ ! = ! ! − ! ! ! ! − ! ! !" !" = 2 ! ! = ! ! ! = 2 ! ! ! ! = 2 ! 0 0 0 0 ! = ! ! !" !" = 3 ! ! ! = ! ! = 1 !" !" = 1 ! = 3 ! ! ! ! = 3 ! ! ! = 1 ! ! ! ! = 1 Cubic Quadratic Line (y = mx + b) Constant

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10 Derivative relationships • Sinusoids ! = sin ! !" !" = cos ! 0 0 0 0 ! = ! ! ! = ! ! ! = ! ! = !"#$%&#% Derivative relationships 0.00 0.50 1.00 1.50 Hip height (m) -1.00 0.00 1.00 2.00 3.00 Hip velocity (m/s) -15.00 -10.00 -5.00 0.00 5.00 10.00 15.00 Hip acceleration (m/s 2 ) Time (s) position vs. time → velocity vs. time → acceleration vs. time Inflection point → maximum/minimum → zero