Understanding Kinematics in Motion Analysis

1. Chapter 2 Kinematics in One Dimension

2. Analyzing Motion

3. Displacement The displacement of an object is a vector that points from an object initial position to its final position and has a magnitude that equals the shortest distance between the two positions. ∆ SI Unit: meter (m) When motion is along a straight line, directions can be assigned as positive or negative

4. More Displacement The displacement is the difference between the final and initial position of an object. Remember that delta means change in. The change in any variable is always the final value minus the initial value.

5. Speed and Velocity Speed DOES NOT tell you anything about direction. • Speed can be described as how fast an object is moving. • Average speed is the distance traveled divided by the time required to cover the distance. • SI unit: m/s

6. Average Velocity Average velocity is found by dividing the displacement by the elapsed time. SI unit: m/s Average velocity is a vector that points in the same direction as the displacement. Again, for one dimension, + or – can be used to indicate direction of velocity.

7. Instantaneous Velocity The instantaneous velocity of an object indicates how fast the object moves and the direction of the motion at each instant of time.

8. Questions So Far http://he-cda.wiley.com/WileyCDA/HigherEdMultiTitle.rdr?name=cutnell What is the difference between speed and velocity? Give an example of positive velocity and negative velocity. Explain how average velocity and instantaneous velocity are different.

9. Displacement vs. Time Graphs The slope of a displacement vs. time graph will tell you the velocity of the object. If the slope is positive, the velocity is positive. It should go without saying that if the slope is negative, the velocity is negative

10. Look at a Few Examples Constant VelocityPositive Velocity Positive VelocityChanging Velocity (acceleration) Leftward (-) VelocityFast to Slow Negative (-) VelocitySlow to Fast Fast, Leftward(-)Constant Velocity How could we find the instantaneous velocity in any of these situations?

11. Velocity vs. Time Graphs The slope of a velocity vs. time graph reveals useful information about the acceleration of an object If the acceleration is zero, the slope is zero (horizontal) If acceleration is positive, slope is positive If the acceleration is negative, slope is negative

12. Reading Velocity vs. Time Graphs http://www.physicsclassroom.com/Class/1DKin/u1l4c.cfm

13. The Area Under the Curve of a V-t graph The area under the curve of a velocity vs. time graph can be used to determine the displacement of an object.

14. Acceleration • Acceleration comes into discussions of motion when the change in the velocity is combined with the time during which the change occurs. • Average Acceleration is a vector that points in the same direction as • As with velocity, plus and minus signs indicate the two possible direction for the acceleration vector when motion is along a straight line.

15. Definition of Average Acceleration SI unit of acceleration: m/s2 Instantaneous acceleration is a limiting case of the average acceleration. When the time interval for measuring the acceleration becomes extremely small, the average acceleration and the instantaneous acceleration become equal. In most situation we deal with acceleration is constant or “uniform.”

16. Sample Problem A drag racer crosses the finish line, and the driver deploys a parachute and applies the brakes to slow down. The driver begins slowing down when t = 9.0s and the car’s velocity is +28m/s. When t = 12.0s, the velocity has been reduced to +13m/s. What is the average acceleration of the dragster?

17. Negative and Positive Acceleration Whenever the acceleration and velocity vectors have opposite directions, the object slows down and is said to be “decelerating.” When the acceleration and velocity vectors point in the same direction, the object speeds up and has positive (+) acceleration.

18. Equations of Kinematics for Constant Acceleration • We have now discussed the motion of objects along a straight line in terms of the following quantities • Displacement • Velocity • Time • Acceleration • The kinematics equations use no new concepts but relate these terms in easy to use equations. • These equations need only be derived one time, then used over and over.

19. Derivation Assumptions • Object is located at the origin when . Then, ∆x becomes x. • Dispense using boldface symbols and small arrows for displacement, velocity, and acceleration vectors. • Continue to use + or – for direction. • Essentially, kinematics equations rearrange and recombine existing equations in order to make problem solving easier.

20. Kinematics Equations 1. Start with acceleration equation and rearrange to solve for v. Since the velocity increases at a constant rate, the average velocity is midway between v0 and v. 2. Using the average velocity formula and assuming x0 and t0 to be zero, x = vt .

21. Kinematics Equations Continued 3. Combining equations one and two into a single equation by algebraically eliminating the final velocity (v) expression for displacement (x) will result. 4. Finally, by combining the average acceleration formula (rearranged to solve for t) and equation number 2, an expression of motion not involving time can be derived.

22. Hints for Solving So Far • Draw a Picture • Decide which direction will be positive (+) and which will be negative (-) • While you are reasoning the problem, be sure to interpret the terms “decelerating” or “deceleration” correctly • If the motion of two objects are interrelated, data for only two variable need to be specified for each object. • If motion is divided into segments, with different acceleration, realize that final velocity for one is initial velocity for the other.

23. Freely Falling Bodies In general, all bodies at the same location above the earth fall vertically with the same acceleration. When we neglect air resistance and consider acceleration nearly constant, we are considering “free fall” and can use kinematics equations in problem solving g = 9.8m/s2

24. Using Kinematics Equations for Free Fall g on the moon is approx 1/6 that of g on Earth Often, the symbol “y” is used for vertical displacement When an object is dropped, v0 = 0m/s The acceleration due to gravity is ALWAYS a downward-pointing vector.

25. Free Fall Up? Free Fall refers to any object moving either upward or downward under the influence of gravity alone In either case, the object always experiences the same downward acceleration due to gravity