Chapter 3

1. 0° Kinematics in Two Dimensions Chapter 3

2. Position vector vs. Displacement vector final displacement initial Displacement is the distance and direction that the object moved.

3. Average velocity vectoris the displacement vector divided by the elapsed time. Average velocity gives the velocity for a long time interval Δt. The actual velocity may vary during that time interval.

4. The instantaneous velocity vectoris the average velocity vector calculated for a very short time interval.Instantaneous velocity is the velocity at a particular moment.

5. Instantaneous velocity vector

6. Average acceleration vectoris the change in the instantaneous velocity vector divided by the elapsed time. Direction of the acceleration vector is the same as the direction of the velocity change vector

7. Review Kinematic equation used in ch 2 described one-dimensional (1D) motion along a horizontal or vertical straight line. Horizontal Vertical Two-dimensional (2D) motion has both horizontal and vertical motion.

8. Two-dimensional kinematic equations These equations are used to calculate horizontal components. The x subscripts specify that these are x components.

9. Two-dimensional kinematic equations These equations are used to calculate vertical components. The y subscripts specify that these are y components.

10. The x and y motions are independent of each other Each axis has its own values for position, initial velocity, final velocity, and acceleration.The time is the same for motion along the two axes.

11. Example 1 A Moving Spacecraft In the x direction, the initial velocity component is +22 m/s and an acceleration is +24 m/s2. In the y direction, the initial velocity component is +14 m/s and an acceleration is +12 m/s2. At time 7 s, calculate (a) x and vx (b) y and vy (c) the final velocity vector First: Success Strategy Then: Apply it to this example.

12. Success Strategy 1. Make a drawing.2. Draw direction arrows. (displacement, velocity, acceleration) 3. Label positive (+) and negative (-) x and y directions.4. Label all known quantities.5. Use a table to organize known values with correct units. 6. You need three of the five kinematic quantities for y motion. 7. You need three of the five kinematic quantities for x motion. 8. Time is the same quantity for x and y motion. 9. Select the appropriate equations and solve. When the motion is divided into segments, remember that the final velocity of one segment is the initial velocity for the next. There may be two solutions for some quantities because quadratic functions have two roots. Use reasoning to select the correct one.

13. Example 1 A Moving Spacecraft In the x direction, the initial velocity component is +22 m/s and an acceleration is +24 m/s2. In the y direction, the initial velocity component is +14 m/s and an acceleration is +12 m/s2. At time 7 s, find (a) x and vx, (b) y and vy, and (c) the final velocity of the spacecraft.

14. Solution for the x components

15. Solution for the y components

16. vx = +190 m/s vy = +98 m/s x = +742 m y = +392 m

17. Find the spacecraft's final velocity vector .

18. Projectile motion Projectile motion is 2D motion. We will only consider free-fall motion that is only influence by gravity. We will ignore air resistance.Near the surface of the Earth the acceleration due to gravity is 9.8 m/s2 downward. Therefore vx always stays the same

19. Example 3 A Falling Care Package An airplane moving horizontally with a constant velocity of +115 m/s at an altitude of 1050 m releases a care package. How long does it take for the package to hit the ground?What is the final velocity vector just before it hits the ground? vox = +115 m/s voy = 0 y = 0 0° y = -1050 m

20. vox = +115 m/s voy = 0 y = 0 y = -1050 m

22. The horizontal and vertical velocity components are independent of each other. The initial vertical velocity was zero just as if the package were dropped from rest. The package falls with the same vertical velocities as a package that had been dropped from rest.

23. 0° -51.2°

24. Conceptual Example 5 Bullet shot into the air... Suppose you are driving a convertible with the top down. The car is moving to the right at constant velocity. You point a rifle straight up into the air and fire it. In the absence of air resistance, where would the bullet land – behind you, ahead of you, or in the barrel of the rifle? ???? What principles have we learned that can guide your answer?

25. Example 6 Height of a Kickoff A player kicks a football with an initial speed of 22 m/s at a 40°angle. How high will the ball go? (In this course, we always ignore air resistance.) 0° 40° What unique characteristic does the “top” of the arc have? Answer: the vertical velocity component vy at the top is zero.

26. Find the x and y components of the initial velocity vector

27. ytop = ?? 0° At the top

29. Example 7 The Time of Flight of a Kickoff What is the time of flight between kickoff and landing? What is the y position where the ball hits the ground?

30. y = 0 y = 0

31. two solutions time for y = 0 at kick time for y = 0 at landing

32. Example 8 Calculate the range R of the projectile. Range R is the x coordinate where the ball hits the ground. X=? =

33. The End