Slide 1 : Speed is a scalar.
Velocity is a vector. Velocity and Speed
Slide 2 : The most common unit of velocity is meters per second (m/s). Velocity and Speed
Slide 3 : Velocity and Speed
Slide 4 : Velocity and Speed
Slope FIGURE 8-11 A runner moves along the path followed by the dotted line. The length of this path is the distance traveled. The length of the solid line is the displacement.
Slide 5 : Velocity and Speed
Slope FIGURE 8-14 Different slopes on a vertical position versus time graph. Slopes a, b, and e are positive. Slopes c and f are negative; d has zero slope.
Slide 6 : Velocity and Speed
First central distance method FIGURE 8-15 The location in time of velocity. A. Using the traditional method over a single time interval. B. Using the first central difference method.
Slide 7 : Velocity and Speed
Numerical example
Slide 8 : Velocity and Speed
Numerical example FIGURE 8-16 Position-time profile (A) and velocity-time profile (B) of the data in Table 8-2.
Slide 9 : Velocity and Speed
Instantaneous velocity FIGURE 8-17 The slope of the secant a is the average velocity over the time interval t1 to t4. The slope of secant b is the average velocity over the time interval t2 to t3. The slope of the tangent is the instantaneous velocity at the time interval ti when the time interval is so small that in effect it is zero.
Slide 10 : Velocity and Speed
Graphic Example FIGURE 8-18 Local extrema (slope 0) on a position-time graph.
Slide 11 : Velocity and Speed
Graphic Example FIGURE 8-19 The position-time curve (A) and the respective velocity-time curve (B) drawn using the concepts of local extrema and slopes.
Slide 12 : Acceleration is used in everyday terms as a scalar.
It is, strictly speaking, a vector. Acceleration
Slide 13 : The most common unit of acceleration is meters per second squared (m/s2). Acceleration
Slide 14 : Acceleration
Instantaneous Acceleration
Acceleration and the direction of motion FIGURE 8-20 Motion to the right is regarded as positive and to the left is negative. Positive or negative velocity is based on the direction of motion. Acceleration may be positive, negative, or zero based on the change in velocity.
Slide 15 : Acceleration
Acceleration ad the direction of motion FIGURE 8-21 The graphical relationship between acceleration and direction of motion during a shuttle run (t2 denotes when the runner changed direction).
Slide 16 : Acceleration
Numerical Example
Slide 17 : Acceleration
Numerical Example FIGURE 8-22 Velocity-time profile (A) and acceleration-time profile (B) for Table 8-3
Slide 18 : Acceleration
Graphical Example FIGURE 8-23 The relationship between the velocity-time curve and the acceleration-time curve drawn using the concepts of local extrema and slopes.
Slide 19 : Differentiating positional data to get velocity and acceleration has been covered.
However, acceleration may be collected in a biomechanical analysis.
In this case, you may want to calculate velocity and displacement form acceleration data.
This is the opposite of differentiation and is known as integration. Differentiation and Integration
Slide 20 : Finite differentiation methods are used with digital data.
Similarly, finite integration methods are used with digital data.
Finite differentiation calculates the slope of the curve.
Finite integration calculates the area under the curve. Differentiation and Integration
Slide 21 : If:
Then:
Hence:
Area under an acceleration time curve = a?t = velocity Differentiation and Integration
Slide 22 : Differentiation and Integration FIGURE 8-24 An idealized acceleration-time curve. Area A equals (3 m/s2 * 6) s or 18 m/s. This represents the change in velocity over the time interval from 0 to 6 s. The change in velocity for area B is 14 m/s.
Slide 23 : Finite differentiation approximates the area under curves as a series of rectangles.
This is called the Riemann sum.
If ?t is small enough, this is an accurate approximation. Example above: Horizontal velocity time curve with 30 time intervals. Integral equals change in displacement. Differentiation and Integration ds => ?s
“d” => delta => ?