# LINEAR AND ANGULAR KINEMATICS Readings: McGinnis Chapters 2 and 6 DISTANCE, DISPLACEMENT, SPEED, VELOCITY, AND ACCELERATION:

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1 LINEAR AND ANGULAR KINEMATICS Readings: McGinnis Chapters 2 and 6 1 DISTANCE, DISPLACEMENT, SPEED, VELOCITY, AND ACCELERATION: How far? Describing change in linear or angular position Distance (Scalar quantity) Length of path traveled with no concern for direction (magnitude only) Displacement (Vector quantity) Difference between starting and finishing positions; independent of path followed (magnitude and direction) Symbols: linear - l, d angular - φ, θ Examples - Linear distance and displacement Describing race distances: 100 m Sprint l = 100 m Indy 500 (auto race) l = 500 mi

2 4000 km Tour de France (bike race) l = 4000 km Characterizing performance: Shot put, Long jump, High jump long jump of 8.89 m l = 8.89 m 2 Typical units of measurement: m, km, ft, mile Distinguishing distance and displacement Example: 400 m run (one lap around track) Distance? Displacement? 400 m

3 3 Example: Walking north and east, then north and east again: d 70 m 0 Start 100 m Overall distance traveled? l Displacement ( as the crow flies )? Magnitude d Direction θ

4 Examples - Angular distance and displacement. 4 Diving, gymnastics: triple somersault with a full twist 3.0 revolutions about a transverse axis (somersaulting axis) combined with 1 rev about a longitudinal axis (twisting axis) transverse axis φ = θ= long. axis φ = θ= Discus throw: Body rotates through rotations prior to discus release. φ = rev θ = rev Joint range of motion: The elbow and knee have ROM's of approximately 150 degrees. φ = θ = Typical units - 3 used commonly: revolutions, radians, degrees 1 rev = 2π rad (i.e., 6.28 rad) = 360 deg 1 rad = 57.3 deg

5 How fast? 5 Describing rate of change of linear or angular position with respect to (wrt) time Speed or Velocity: Rate at which a body moves from one position to another Speed (scalar) Velocity (vector) Note: There will be times when it is important to use velocity because of its vector characteristic. Linear: Angular: Examples of linear speed or velocity: Tennis: 125 mph (55.9 m/s) serve v = 125 mph = 55.9 m/s Pitching: 90 mph (40.2 m/s) fastball v = 90 mph = 40.2 m/s

6 Running: Marathon (26.2 mi) - 2 hr 10 min (2.17 hr) v = 12.1 mph (5.4 m/s; 4:57 per mile) Sprinting m in 9.86 s v = m/s = 22.7 mph Football 4.6 speed (40 yd in 4.6 s) v = 7.95 m/s = 17.8 mph 6 Typical units of measurement for v: m/s, km/hr (kph), ft/s, mph Examples of angular speed or velocity: Auto engine speed rpm ω = 3000 rpm CD ROM drive rpm ω = 400 rpm Cycling cadence - 90 rpm ω = 90 rpm Isokinetic dynamometer (Cybex and others) - e.g., 60 deg/s ω = 60 deg/s

7 Body segment ω s - e.g., peak ω of soccer player s knee deg/s. 6.7 rev/s 7 Typical units of measurement for ω: rpm, deg/s, rad/s Average vs. Instantaneous Velocities Average speed per race distance: 100 m...marathon Race that produces highest average speed?

8 Sometimes v over relatively long t is not very informative - reflects need for a kinematic profile for more detailed information about performance e.g., distance running - split times 8 e.g., sprinting T&F World Champ. Johnson vs. Lewis 9.83 s 9.93 s = s Where was race won or lost? Reaction time: = s After 10 m: = s By decreasing t over which we examine velocity and other kinematic information, we approach instantaneous estimates of performance - gives more detail about performance. e.g., during a soccer kick (next page):

9 9 Acceleration Describing the rate of change of linear or angular velocity wrt time Vector only - no scalar equivalent Linear: Angular:

10 Typical units of measurement for acceleration: Linear - m/s 2, ft/s 2 Angular - deg/s 2, rad/s 2 10 Example - linear An athlete described as being quick or having good acceleration - same as being fast? No, acceleration describes a person s ability to change speed or direction; doesn t describe top speed. Example - angular Throwing a baseball Angular speed of shoulder internal rotation increases from zero to 1800 deg/s in 26 ms just prior to release... Ball velocity at release correlates strongly (r=.75) with shoulder internal rotation speed at release (Sherwood, 1995).

11 Example linear: Auto accelerations - Indy car vs. Corvette 11

12 When is speed or velocity highest for Indy car? 12 When is acceleration highest for Indy car? Is the speed or velocity of the Indy car changing approximately 6-8 seconds after the start? Link between position, velocity, and acceleration: Velocity - rate of change of position wrt time Acceleration - rate of change of velocity wrt time Instantaneous velocity is reflected by the slope of the position curve at some instant in time. Instantaneous acceleration is reflected by the slope of the velocity curve at some instant in time. See soccer kick - angular position, velocity, and acceleration graphs next page

13 Knee angular kinematic profile (soccer kick) 13

14 100 m dash linear kinematic profile (highly skilled sprinter) 14 When is maximum velocity achieved? When is acceleration highest? What is the approximate acceleration of sprinter 6 seconds into the race?

15 Position Velocity Time 15 Can you deduce the position vs. time and acceleration vs. time profiles based on the velocity profile of a hand reaching for a glass of water? Acceleration Time Time

16 16 Six Cases of Acceleration Case Change in Speed Starting Direction Ending Direction v 1 Speeding up 2 Slowing down 3 Speeding up 4 Slowing down 5 Reversing directions 6 Reversing directions Notes: 1. Speeding up in the positive direction = acceleration 2. Slowing down in the positive direction = acceleration 3. Speeding up in the negative direction = acceleration 4. Slowing down in the negative direction = acceleration Accel. ( v/ t) 5. Reversing directions from positive to negative = acceleration 6. Reversing directions from negative to positive = acceleration.

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