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Linear Kinematics

Linear Kinematics. Chapter 3. Definition of Kinematics. Kinematics is the description of motion. Motion is described using position, velocity and acceleration. Position, velocity and acceleration are all vector quantities. . Velocity.

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Linear Kinematics

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  1. Linear Kinematics Chapter 3

  2. Definition of Kinematics • Kinematics is the description of motion. Motion is described using position, velocity and acceleration. • Position, velocity and acceleration are all vector quantities.

  3. Velocity • Velocity is defined as the rate of change in position, or the slope of the position – time graph. The units for velocity are m/s.

  4. Relationship between Slope and Velocity

  5. If the slope is horizontal the velocity must be zero. • If the slope is upward the velocity must be positive. • If the slope is downward the velocity must be negative. • Notice that point 1 has less slope than point 2 & 3, compare there velocities.

  6. Taking a Derivative • The process of evaluating the slope to get the rate of change is called taking a derivative. • The rules for estimating velocity from position are: If the slope is horizontal, the velocity is 0. If the slope is positive (up), the velocity is positive. If the slope is negative (down), the velocity is negative.

  7. Acceleration • Acceleration is defined as the rate of change in velocity, or the slope of the velocity – time graph. • The units for acceleration are m/s2.

  8. If the slope of velocity is horizontal the acceleration must be zero. • If the slope of velocity is upward the acceleration must be positive. • If the slope of velocity is downward the acceleration must be negative.

  9. Acceleration is the Derivative of Velocity • The rules for estimating acceleration from velocity are: If the slope of velocity is horizontal, the acceleration is 0. If the slope of velocity is positive (up), the acceleration is positive. If the slope of velocity is negative (down), the acceleration is negative.

  10. Integration • Integration is the mathematical process of getting the area underneath a curve. • Integration of acceleration gives the change in velocity. • Integration of velocity gives the change in position. • The integral sign can be interpreted as get the area underneath the curve. ∫ The change in velocity over the interval from t0 to t1 is equal to the area underneath the acceleration – time curve.

  11. Integration of Acceleration There are several methods of integration. Determining the area of a rectangle is one method of integration. Area = (Height × Width) + Initial Value V = Height x Width V = (2 m/s2)(2 s) V = 4 m/s Area = 4 m/s, velocity changes by 4 m/s. Over the interval from t = 0 to t = 2 s the velocity must change by +4 m/s.

  12. Integration of Acceleration V = Height x Width V = (−3 m/s2)(3 s) V = −9 m/s Area = 9 m/s, velocity changes by 9 m/s. Over the interval from t = 2 to t = 5 s the velocity must change by −9 m/s.

  13. Integration of Acceleration V = Height x Width V = (4 m/s2)(2 s) V = 8 m/s Area = 8 m/s, velocity changes by 8 m/s. Over the interval from t = 5 to t = 7 s the velocity must change by +8 m/s.

  14. Integration of Velocity The integration of velocity gives the change in position. P = Height x Width P = (−3 m/s)(2 s) P = −6 m Area = −6 m, position changes by −6 m. Over the interval from t = 0 to t = 2 s the position must change by −6 m.

  15. Integration of Velocity The integration of velocity gives the change in position. P = Height x Width P = (2 m/s)(3 s) P = +6 m Area = +6 m, position changes by +6 m. Over the interval from t = 2 to t = 5 s the position must change by +6 m.

  16. Integration of Velocity The integration of velocity gives the change in position. P = Height x Width P = (−4 m/s)(3 s) P = −12 m Area = −12 m, position changes by −12 m. Over the interval from t = 5 to t = 8 s the position must change by −12 m.

  17. Evaluate slope to estimate velocity Evaluate area to estimate position A zero for velocity is a local max or min in position

  18. Computing Velocity from Position in Excel =(B3 − B2)/0.1 Excel Filename: Get Vel & Accel Data Set 1.xls

  19. Computing Acceleration from Velocity in Excel =(C4 − C3)/0.1 Excel Filename: Get Vel & Accel Data Set 1.xls

  20. Integration of Acceleration in Excel Velocity Final = (Acceleration Time) + Velocity Initial The general equation for integration is: Area = (Height Width) + Initial Value

  21. Integration of Acceleration in Excel =(D4 * 0.1) + E3 Area = (Height Width) + Initial Value

  22. Integration of Velocity in Excel Position Final = (Velocity Time) + Position Initial The general equation for integration is: Area = (Height Width) + Initial Value

  23. Integration of Velocity in Excel =(E3 * 0.1) + F2 Area = (Height Width) + Initial Value

  24. Relationship between Acceleration & Velocity

  25. Relationship between Velocity & Position

  26. What Does The Initial Value Do? Area = (Height Width) + Initial Value The initial value tells you where to start. It simply moves the curve up or down on the Y axis.

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