Fundamentals of Kinematics for Motion Analysis

1. Kinematics Velocity and Acceleration

2. Motion • Change in position of object in relation to things that are considered stationary • Usually earth is considered stationary • Nothing is truly stationary (earth travels 108,000 km/hr orbiting sun) • All motion is relative: must be related to other objects called your frame of reference

3. Distance and Displacement • Distance: how far object moves without respect to direction, a scalar quantity • Displacement: change of position in a particular direction; How far and in what direction object is from original position, a vector • Both use symbol d ,and often x or y for 1-dim motion • The unit is the meter

4. Speed • The time rate of motion, the rate of change of position--a scalar • Units of distance /time: m/s usually but can be miles/hr, km/hr; Symbol is v • Average speed = total distance/elapsed time • Instantaneous speed: rate of change of position at any instant

5. Velocity • Speed in a particular direction, a vector; Unit same as speed; Symbol v • Must include a direction, using angle from known reference points, compass headings, or just left & right, + & -,up & down • Can be negative (going backwards) • Average velocity = total displacement / elapsed time

6. Velocity • Instantaneous velocity: instantaneous speed with current direction • Constant velocity means no change of speed or direction • Often we are interested in only the speed (we may know the direction) so speed and velocity are sometimes used interchangeably

7. Acceleration • Time rate of change of velocity; A vector; Symbol a and units of m/s/s usually shortened to m/s2 • Acceleration can be negative • Average acceleration = change in velocity / elapsed time for the change • Galileo first to understand acceleration

8. 1st Constant Accel. Equation • If acceleration is constant, instantaneous acceleration always equals avg acceleration • Use definitions of avg velocity and accel to calculate final velocity or distance • Since a = (vf - vi)/t , then vf = vi + at • If vi = 0 , then vf = at • Use when distance not given or asked for

9. 2nd Constant Acceleration Equation • vavg = (vf + vi)/2 ; but also vavg = d/t ; so (vf + vi)/2 = d/t • Now using our first equation for vfwe can get (vi + vi + at)/2 = d/t • Solving for d: d = vit + 1/2 at2 • If vi = 0, d = 1/2 at2 • Use when final speed not given or asked for

10. 3rd Constant Acceleration Equation • Solve 1st equation for t and substitute into 2nd equation, expand squared quantity and combine terms. • Get 2ad = vf2- vi2; solve for vf2 • vf2= vi2 + 2ad • If vi = 0, vf2= 2ad • Use when time is not given or asked for

11. Graphing Motion: d vs t • Plot time as independent variable • On position vs time graph, slope at any value of t gives instantaneous velocity • If graph is linear, slope and v are constant • If graph is curved, slope and v are found by drawing tangent line to curve and finding its slope

12. Graphing Motion: d vs t • Uniform motion (constant velocity)

13. Graphing Motion: d vs t (x vs t) • Accelerated motion (increasing velocity)

14. Graphing Motion: v vs t • Slope of v vs t graph gives acceleration • If graph is linear, acceleration is constant • If graph is curved, instantaneous acceleration is found using slope of tangent line at any point

15. Tangent Line A line that just touches a curve at one point and gives the slope of the curve at that point.

16. Velocity vs Time: acceleration

17. Comparing Uniform and Accelerated Motion Graphs Uniform motion Accelerated Motion

18. Uniform motion Accelerated Motion Comparing Uniform and Accelerated Motion Graphs

19. Comparing Positive and Negative Velocity

20. Speeding up and Slowing Down

21. Velocity vs Time Graphs: Finding Displacement • Displacement can be found from velocity graph by finding the area between the line of the graph and the time axis • Divide the area bounded by the graph line, the horizontal axis and the initial and final times into geometric sections (squares, rectangles, triangles, trapezoids) and find the area • Area below the time axis is negative displacement

22. Area under (enclosed by) the Velocity Graph

23. Area Enclosed by the Velocity Graph • Divide complex areas into triangles and rectangles

24. Area Enclosed by the Acceleration Graph • If acceleration vs. time is plotted, area between the graph line and the horizontal (time) axis gives the change in velocity that took place during the time interval

25. Free Fall • Common situation for constant acceleration is free fall • Force of gravity causes falling bodies to accelerate • Force varies slightly from place to place but average acceleration is 9.80 m/s2 designated by symbol g • Often for simplicity or approximations, g = 10 m/s2 is used

26. Free Fall • Distance increases with each second of falling. • Object will fall 4.9 m (about 5 m) during the 1st second • Distance increases by 9.8m (about 10 m) each second • Speed increases by 9.8 m/s (about 10 m/s) for each second of falling

27. Keeping Track of the Signs • If motion is only in one direction (usually down), using positive and negative signs to indicate direction is not necessary. • With up and down motion, up is considered positive and down negative • g must be negative (-9.80 m/s2) in these situations along with downward displacements and velocities

28. Air Resistance and Free Fall • If air drag is ignored, all objects fall at the same rate • Air resistance slows rate of fall, depending on object’s surface area, shape, texture and density of air • For our purposes, air resistance is negligible

29. Equations for Free Fall • Can use all constant acceleration equations for free fall • Equations for vertical motion are written with symbol g in place of a and y in place of x or d • Be careful with positive and negative signs!

30. Horizontal Motion Vertical Motion Constant Acceleration Equations