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# 2. linear kinematics i

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• 1. LINEAR KINEMATICSKIN3323: Biomechanics
• 2. TYPES OF MOTION
• 3. Kinematics Description of the form, pattern, or sequencing of movement with respect to time –  Involves: position, velocity and acceleration of a body without concern for the forces which cause motion –  Does not involve force
• 4. Types of Motion 1. Linear Motion –  Translation: all points on an object move the same distance, in the same direction, and at the same time’ –  2 Types a) Rectilinear: Movement in straight line path b) Curvilinear: Movement in straight curved path
• 5. 1a. Rectilinear Translation (Linear Motion)Example •  Figure skater gliding across the ice in a static position
• 6. 1a. Rectilinear Translation (Linear Motion)Example •  Figure skater gliding across the ice in a static position
• 7. 1b. Curvilinear Translation (Linear Motion)Example •  Skateboarder in air holding a static position •  Ski jumper
• 8. Types of Motion 2. Angular Motion –  Rotation: all points on an object moves in circles about the same fixed axis
• 9. 2. Angular MotionExample •  Elbow flexion •  Figure skater spinning
• 10. Types of Motion 3. General Motion –  Combination of translation and rotation –  Most common type of motion in sports and human movement
• 11. DESCRIPTION OF MOTION
• 12. Description of motion •  Position •  Distance / displacement •  Speed / velocity •  Acceleration
• 13. Description of motion: Position •  The location of a point, with respect to the origin, within a spatial reference frame •  Reference frames –  Cartesian coordinate system •  Fixed point (origin) with axes that are perpendicular to each other •  2-D or 3-D system
• 14. 2-D Coordinate System Used when motion is +y primarily in one plane •  x (horizontal) •  y (vertical)   -x +x Origin = (0,0) (0,0) means that the point is located at x=0 & y=0   ex. (2,5) describes the -y point located at x=2 & y=5
• 15. Example of reference frames in Biomechanics +y +y -x +x -x +x Origin Origin -y -y Global reference frame Anatomical reference frame Relative to gravity Relative to body segment (ex. forearm relative to arm)
• 16. Distance vs. Displacement Distance Displacement •  Length of path •  Straight line distance followed by object in a specific direction from initial to final from initial to final position position •  Scalar quantity •  Vector quantity –  Only magnitude –  Has magnitude and direction
• 17. Distance = length of path y Distance  =  1+1.5+3+4+1+2+3.8+2  =  17.8   12   11   1.5m   10   1m   3m   2m   9   (11,10) 3.8m   8   7   (2,7) 6   5   4m   2m   4   3   1m   2   1   x (0,0) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
• 18. Displacement = Straight line distance y 12   11   10   9   (11,10) Δy 8   7   (2,7) Δx 6   5   4   3   2   1   x (0,0) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
• 19. Displacement = Straight line distance (11,10) Δy = 10 – 7 = 3 (2,7) Δx = 11 – 2 = 9 Displacement2 = Δx2 + Δy2 Displacement = Δx2 + Δy2 = 92 + 32 = 9.5
• 20. Which is more relevant: Distance or displacement? 400m  Race
• 21. Which is more relevant: Distance or displacement? 400m  Race   Displacement?   Distance?   Start   End
• 22. Which is more relevant: Distance or displacement? Javelin  throw
• 23. Which is more relevant: Distance or displacement? Distance  =  200m   Distance  =  200m   Displacement   Displacement   =  20m   =  80m   Javelin  throw
• 24. Description of motion: Speed vs. Velocity Speed Velocity •  A distance traveled •  A displacement over time achieved over time •  Scalar quantity •  Vector quantity –  Magnitude only –  Has magnitude and direction
• 25. Speed vs. Velocity 10 20 30 40 50 40 30 20 10 10 20 30 40 50 40 30 20 10 A football player caught a ball at 0 yd line. He ran 52 yd and got tackled after gaining 40 yds and moving 20 yds to the left. The play lasted 5 sec.
• 26. Speed = distance traveled over time Speed = distance / Δ time Distance = 55 yd ΔTime = 5 sec Speed =  distance / Δ time = 55 yd / 5 sec He ran 55 yds over 5 sec = 11 yd / sec
• 27. Velocity = displacement achieved over time Velocity = displacement / Δ time Displacement = 402 + 202 = 44.7 yd ΔTime = 5 sec 20yds Speed =  distance / Δ time = 44.7 yd / 5 sec 40yds = 8.9 yd / sec
• 28. Velocity = displacement achieved over time Alternative solution 20yd/5sec = 4yr/sec 40yd/5sec = 8yr/sec Velocity = (8yr/sec)2 + (4yr/sec)2 = 8.9 yd / sec
• 29. Average vs. instantaneous speed / velocity Average Instantaneous •  Speed or velocity •  Speed or velocity at a averaged over time specific instant –  ex. Running pace –  ex. Speedometer reading •  Used when interested “10 minute mile” in knowing = 1mile / 10min –  Max/min values = 0.1 mile / min –  Values at specific instant = 6 mile / hour = 6 mph
• 30. Instantaneous velocity At take off, the high jumper’s vertical velocity was 9m/s and her horizontal velocity was 2m/s. Calculate the take off velocity and angle. ? m/s 9m/s ?° 2m/s
• 31. Instantaneous velocity At the instant of ball release, instantaneous velocity of the ball was 90 mph at a direction 10° above horizontal. What was the vertical and horizontal velocity of the ball? Vy 90mph 10° Vx
• 32. Description of motion: Acceleration •  Rate of change in velocity –  ex. “0 to 60mph in 3 seconds” –  Vector quantity
• 33. Acceleration = Rate of change in velocity Acceleration = Δ velocity / Δ time = (Vf – Vi) / Δ time Where: Vf = final velocity Vi = initial velocity
• 34. Acceleration = Rate of change in velocity Sean is running a 100m dash. When the starter’s pistol fires, he leaves the starting block and continues speeding up until he reaches his top speed of 11m/s 6s into the race. He holds this speed for 2s and then gradually slows down until he crosses the finish line at 9m/s. What was Sean’s acceleration during: -  First 6s of the race -  From 6-8s into the race -  Last 3s
• 35. Acceleration = Rate of change in velocity First 6s: Vi = 0m/s Vf = 11m/s 6-8s: Vi = 11m/s Vf = 11m/s Last 3s: Vi = 11m/s Vf = 9m/s
• 36. (+) and (-) Acceleration •  Positive acceleration indicates that the object is speeding up –  Acceleration •  Negative acceleration indicates that the object is slowing down –  Deceleration
• 37. Acceleration due to gravity •  The rate of change in velocity caused by the force of gravity –  9.81m/s2 downward
• 38. Summary of Kinematic Descriptors Scalar   Vector   Distance   Displacement   “Length  of  path”   “Straight  line  distance”   Speed   Velocity   “Distance  over  Fme”   “Displacement  over  Fme”   Accelera=on   “Change  in  velocity  over  Fme”   “A  rate  of  change  in  velocity”
• 39. CHARACTERISTICS OFPROJECTILE MOTION
• 40. Projectile Motion A rifle is shot in a perfectly horizontal plane. At the same instant a bullet is dropped from the same height. Which bullet hits the ground first, the one shot from the rifle or the one dropped next to the rifle? Don’t  say  it,  think  about  it....
• 41. Projectile Motion •  Projectile is an object that has been projected into the air or dropped and is only acted on by the forces of gravity and air resistance –  In this unit, we consider air resistance negligible •  Examples: –  Soccer ball after impact –  Diver, long jumper, and high jumper after a take off –  Ball dropped from a top of the building
• 42. Projectile Motion Apex Trajectory (Parabolic) Height Distance Release Landing
• 43. Effects of gravity on projectile motion •  The only force acting on projectiles is the gravitational force (ignoring air resistance) •  Gravitational force only affects vertical velocity –  Vertical and horizontal velocity are independent!!
• 44. Effects of gravity on vertical velocity •  Gravity causes 9.81m/s2 acceleration in vertical (downward) direction –  Vertical velocity of the projectile decrease by 9.81m/s every second
• 45. Projectile Motion Apex Height (m) 0 1 2 3 4 5 6 7 839.2m/s   -­‐9.81   -­‐9.81   Velocity (m/s) -­‐9.81   -­‐9.81   0   -­‐9.81   -­‐9.81   -­‐9.81   -­‐9.81   Vertical Distance (m) velocity
• 46. Effects of gravity on horizontal velocity •  Gravity does not cause acceleration in horizontal direction –  Gravity has no influence on horizontal velocity –  Horizontal velocity of the projectile does not change (= stays constant)
• 47. Projectile Motion Apex Position (m) 0 1 2 3 4 5 6 7 8 20.0m/s   Horizontal Velocity (m/s) velocity 0   Time (sec)
• 48. Summary of the characteristics of projectile motion •  Trajectory of the center of mass (COM) of the projectile is parabolic –  Symmetric about the apex –  Time up = time down •  Vertical velocity –  Decreases by 9.81m/s every second during up phase –  0m/s at apex •  Horizontal velocity –  Is constant (ignoring air resistance)
• 49. FACTORS INFLUENCING PROJECTILE MOTION
• 50. Horizontal and vertical velocity •  For the analysis of projectile motion, velocity is often resolved into horizontal (Vx) and vertical (Vy) component Velocity   Vy   θ   Vx
• 51. Effects of projection angle on horizontal and verticalvelocity •  Increasing projection angle will: –  Decrease horizontal velocity –  Increase vertical velocity Smaller  projec0on  angle  =     20m/s   Smaller  Vy  and  greater  Vx     20m/s   18.8m/s   10.0m/s  Greater  projec0on  angle  =     θ=70°   θ=30°  greater  Vy  and  smaller  Vx     6.8m/s   17.3m/s   Changing  projecFon  angle  will  change  the  raFo   between  horizontal  and  verFcal  velocity
• 52. Effects of projection angle on horizontal and verticalvelocity •  Projection speed influences the shape of projectile’s trajectory Steeper projection angle Milder projection angle = taller parabola = flatter parabola
• 53. Effects of projection speed on horizontal and verticalvelocity •  Increasing projection speed will proportionally increase horizontal and vertical velocity 20m/s   14.1m/s   10m/s   7.1m/s   θ=45°   θ=45°   7.1m/s   14.1m/s   Increasing  projecFon  speed  will  proporFonally   increase  both  horizontal  and  verFcal  velocity
• 54. Effects of projection speed on horizontal and verticalvelocity •  Projection speed influences the size of projectile’s trajectory Smaller projection speed Greater projection speed = smaller parabola = greater parabola
• 55. Vertical and horizontal velocity •  To increase vertical velocity, you can: 1.  Increase projection speed 2.  Increase projection angle •  To increase horizontal velocity, you can: 1.  Increase projection speed 2.  Decrease projection angle
• 56. Variables of interest in projectile motion 1. Maximum height •  How high does the object travel? 2. Flight time •  How long does the object stay in air? 3. Flight distance •  How far does the object travel?
• 57. Factors Influencing Maximum Height •  Maximum height is affected by 2 factors –  Vertical velocity –  Projection height Height (m) 0 0 1 2 3 4 5 6 7 8 Distance (m) Increasing  verFcal  velocity  by  increasing  projec=on  velocity  will   increase  maximum  height
• 58. Factors Influencing Maximum Height •  Maximum height is affected by 2 factors –  Vertical velocity –  Projection height Height (m) 0 0 1 2 3 4 5 6 7 8 Distance (m) Increasing  verFcal  velocity  by  increasing  projec=on  angle  will  increase   maximum  height
• 59. Factors Influencing Maximum Height •  Maximum height is affected by 2 factors –  Vertical velocity –  Projection height Height (m) 0 0 1 2 3 4 5 6 7 8 Distance (m) Increasing  projecFon  height  will  increase  maximum  height
• 60. Variables of interest in projectile motion 1. Maximum height •  How high does the object travel? 2. Flight time •  How long does the object stay in air? 3. Flight distance •  How far does the object travel?
• 61. Factors Influencing Flight Time •  Flight time is affected by 2 factors –  Relative height of release (= final height – initial height) •  Difference in height between the time of release and landing –  Vertical velocity Shorter flight time Longer flight time PosiFve  relaFve  height  will   NegaFve  relaFve  height  will   decrease  the  ﬂight  Fme   increase  the  ﬂight  Fme
• 62. Factors Influencing Flight Time •  Flight time is affected by 2 factors –  Relative height of release (= final height – initial height) •  Difference in height between the time of release and landing –  Vertical velocity RelaFve  height  only  aﬀects  0me  down
• 63. Factors Influencing Flight Time •  Flight time is affected by 2 factors –  Relative height of release (= final height – initial height) •  Difference in height between the time of release and landing –  Vertical velocity Height (m) 0 0 1 2 3 4 5 6 7 8 Distance (sec) Increasing  verFcal  velocity  by  increasing  projec=on  velocity  will   increase  ﬂight  Fme
• 64. Factors Influencing Flight Time •  Flight time is affected by 2 factors –  Relative height of release (= final height – initial height) •  Difference in height between the time of release and landing –  Vertical velocity Height (m) 0 0 1 2 3 4 5 6 7 8 Distance (sec) Increasing  verFcal  velocity  by  increasing  projec=on  angle  will  increase   ﬂight  Fme
• 65. Variables of interest in projectile motion 1. Maximum height •  How high does the object travel? 2. Flight time •  How long does the object stay in air? 3. Flight distance •  How far does the object travel?
• 66. Factors Influencing Flight Distance •  Flight distance is affected by 2 factors –  Flight time •  Given the horizontal velocity, longer the object is in air, the longer the flight distance –  Horizontal velocity •  Given the flight time, greater the horizontal velocity, the longer the flight distance Distance   Speed  =   Distance  =   Speed  x  Time   Time
• 67. Factors Influencing Flight Distance •  Flight distance is affected by 2 factors –  Flight time •  Given the horizontal velocity, longer the object is in air, the longer the flight distance –  Horizontal velocity •  Given the flight time, greater the horizontal velocity, the longer the flight distance Flight  =me   Horizontal  velocity   Increase  projecFon  speed   Increase  *   Increase   RelaFve  height  of  release   Increase   no  eﬀect   Increase  projecFon  angle   Increase  *   Decrease   Decrease  projecFon  angle   Decrease  **   Increase   *  by  increasing  verFcal  velocity   **  by  decreasing  verFcal  velocity
• 68. Optimal angle of release depends on: •  Goal of the task –  High jump vs. long jump •  Projection height –  Release Ht = Landing Ht è Optimal projection θ = 45 ° •  Ex: kick a ball for max horizontal displacement –  Release Ht > Landing Ht è Optimal projection θ < 45 ° •  Ex: throw a ball for max horizontal displacement –  Release Ht < Landing Ht è Optimal projection θ > 45 ° •  Ex: throw a ball onto elevated surface
• 69. Summary Variable   Determined  by   Increased  horizontal  velocity   Increased  projecFon  speed   Decreased  projecFon  angle   Increased  verFcal  velocity   Increased  projecFon  speed   Increased  projecFon  angle   Increased  maximum  height   Increased  verFcal  velocity   Increased  projecFon  height   Increased  ﬂight  Fme   Increased  verFcal  velocity   Increased  projecFon  height   Decreased  relaFve  height   Increased  ﬂight  distance   Increased  horizontal  velocity   Increased  ﬂight  Fme