3 Motion in Two & Three Dimensions

1. 3 Motion in Two & Three Dimensions • Displacement, velocity, acceleration • Case 1: Projectile Motion • Case 2: Circular Motion • Hk: 51, 55, 69, 77, 85, 91.

2. Position & Displacement Vectors

3. Velocity Vectors

4. Relative Velocity • Examples: • people-mover at airport • airplane flying in wind • passing velocity (difference in velocities) • notation used:velocity “BA” = velocity of B with respect to A

5. Example:

6. Acceleration Vectors

7. Direction of Acceleration • Direction of a = direction of velocity change (by definition) • Examples: rounding a corner, bungee jumper, cannonball (Tipler), Projectile (29, 30 below)

8. 0 • begins when projecting force ends • ends when object hits something • gravity alone acts on object Projectile Motion

9. 0 Horizontal V Constant

10. 0 Two Dimensional Motion (constant acceleration)

11. 0 Range vs. Angle

12. 0 Example 1: Calculate Range (R) vo = 6.00m/s qo = 30° xo = 0, yo = 1.6m; x = R, y = 0

13. 0 Example 1 (cont.) Step 1

14. 0 Quadratic Equation

15. 0 Example 1 (cont.) End of Step 1

16. 0 Example 1 (cont.) Step 2 (ax = 0) “Range” = 4.96m End of Example

17. Circular Motion • Uniform • Non-uniform • Acceleration of Circular Motion

18. Centripetal Acceleration Turning is an acceleration toward center of turn-radius and is called Centripetal Acceleration Centripetal is left/right direction a(centripetal) = v2/r (v = speed, r = radius of turn) Ex. V = 6m/s, r = 4m. a(centripetal) = 6^2/4 = 9 m/s/s 18

19. Tangential Acceleration • Direction = forward along path (speed increasing) • Direction = backward along path (speed decreasing)

20. Total Acceleration • Total acceleration = tangential + centripetal • = forward/backward + left/right • a(total) = dv/dt (F/B) + v2/r (L/R) • Ex. Accelerating out of a turn; 4.0 m/s/s (F) + 3.0 m/s/s (L) • a(total) = 5.0 m/s/s

21. Summary • Two dimensional velocity, acceleration • Projectile motion (downward pointing acceleration) • Circular Motion (acceleration in any direction within plane of motion)

22. Ex. A Plane has an air speed vpa = 75m/s. The wind has a velocity with respect to the ground of vag = 8 m/s @ 330°. The plane’s path is due North relative to ground. a) Draw a vector diagram showing the relationship between the air speed and the ground speed. b) Find the ground speed and the compass heading of the plane. (similar situation)

25. 0 PM Example 2: vo = 6.00m/s qo = 0° xo = 0, yo = 1.6m; x = R, y = 0

26. 0 PM Example 2 (cont.) Step 1

27. 0 PM Example 2 (cont.) Step 2 (ax = 0) “Range” = 3.43m End of Step 2

28. PM Example 2: Speed at Impact

29. v1 0 1. v1 and v2 are located on trajectory. a

30. Q1. Given locate these on the trajectory and form Dv. 0

31. 0 Velocity in Two Dimensions • vavg // Dr • instantaneous “v” is limit of “vavg” as Dt  0

32. 0 Acceleration in Two Dimensions • aavg // Dv • instantaneous “a” is limit of “aavg” as Dt  0

33. Dr ro r 0 Displacement in Two Dimensions

34. v1 1. v1 and v2 are located on trajectory. a

35. Ex. If v1(0.00s) = 12m/s, +60° and v2(0.65s) = 7.223 @ +33.83°, find aave.

36. Q1. Given locate these on the trajectory and form Dv.

37. a v3 Dv v4 Q2. If v3(1.15s) = 6.06m/s, -8.32° and v4(1.60s) = 7.997, -41.389°, write the coordinate-forms of these vectors and calculate the average acceleration.