1. 2009 Physics 2111 Fundamentals of Physics Chapter 3 1 Fundamentals of Physics Chapter 3 Motion in 2 &3 Dimensions 1.Moving in 2 &3 Dimensions 2.Position & Displacement 3.Average & Instantaneous Velocity 4.Average & Instantaneous Acceleration 5.Projectile Motion 6.Projectile Motion Analyzed 7.Uniform Circular Motion 8.Relative Motion Review & Summary Questions Exercises & Problems

2. 2009 Physics 2111 Fundamentals of Physics Chapter 3 2 Position & Displacement Position Vector: A vector from a reference point (aka the origin) to the particle. The vector gives the position of the green ball.

3. 2009 Physics 2111 Fundamentals of Physics Chapter 3 3 Position & Displacement Displacement Vector: Consider a particle moving along a path.

4. 2009 Physics 2111 Fundamentals of Physics Chapter 3 4 Average & Instantaneous Velocity Average Velocity = ratio of the displacement to the time interval: Instantaneous Velocity = derivative of particle position wrt time: A vector in the same direction as the straight line connecting the starting point to the end point. A vector that is tangent to the particle’s path at the particle’s position.

5. 2009 Physics 2111 Fundamentals of Physics Chapter 3 5 Velocity Components

6. 2009 Physics 2111 Fundamentals of Physics Chapter 3 6 Average & Instantaneous Acceleration Acceleration of a particle need not point along the path of the particle. Average Acceleration = ratio of the change in velocity to the time interval: Instantaneous Acceleration = derivative of particle velocity wrt time: If the velocity of a particle changes in either magnitude or direction, the particle is subject to an acceleration.

7. 2009 Physics 2111 Fundamentals of Physics Chapter 3 7 Projectile Motion The particle is launched with initial velocity v o : For a projectile, the motions in the vertical and horizontal planes are independent of each other: horizontal motion with zero acceleration vertical motion with constant downward acceleration 2-Dimensional Motion: a projectile moves in a vertical plane subject to the downward acceleration due to gravity.

8. 2009 Physics 2111 Fundamentals of Physics Chapter 3 8 Projectile Motions

9. 2009 Physics 2111 Fundamentals of Physics Chapter 3 9 Identical Vertical Motion Two balls are released simultaneously The vertical motions are identical.

10. 2009 Physics 2111 Fundamentals of Physics Chapter 3 10 Projectile Motions v x = v 0x v y = - v 0y Equations that describe Projectile Motion

11. 2009 Physics 2111 Fundamentals of Physics Chapter 3 11 Trajectory of a Projectile Particle Trajectory: parabolic path

12. 2009 Physics 2111 Fundamentals of Physics Chapter 3 12 Range of a Projectile

13. 2009 Physics 2111 Fundamentals of Physics Chapter 3 13 Range equation

14. 2009 Physics 2111 Fundamentals of Physics Chapter 3 14 Maximum Range of a Projectile R is a maximum for  0 = 45 o.

15. 2009 Physics 2111 Fundamentals of Physics Chapter 3 15 Range of a Projectile

16. 2009 Physics 2111 Fundamentals of Physics Chapter 3 16 Projectile Motion Applet http://physics.uwstout.edu/physapplets/virginia/www.phys.vir ginia.edu/classes/109n/more_stuff/applets/ProjectileMotion/e napplet.html

17. 2009 Physics 2111 Fundamentals of Physics Chapter 3 17 The Ball Hits the Can Every Time! Magnet (M) releases the can just as the projectile leaves the blow gun (G). During the time-of-flight, both the projectile and the can fall the same distance, h, under the constant acceleration –g.

18. 2009 Physics 2111 Fundamentals of Physics Chapter 3 18 Projectiles

19. 2009 Physics 2111 Fundamentals of Physics Chapter 3 19 Object Falling from a Moving Plane released horizontally h = 500m, v 0 = 55 m/s At what value of  should the payload be released?

20. 2009 Physics 2111 Fundamentals of Physics Chapter 3 20 Cannon Firing at Pirate Ship V 0 cannonball = 82 m/s Pirate ship is 560 m offshore a)What  for cannonball to hit ship b)Safe range for the ship?

21. 2009 Physics 2111 Fundamentals of Physics Chapter 3 21 Great Zacchini The “flight of Emanuel Zacchini” over 3 ferris wheels v 0 = 26.5 m/s,  0 = 53 0, h 0 = 3.0 m Does he clear first Ferris wheel ? If h max is above second Ferris Wheel, by how much does he clear it?

22. 2009 Physics 2111 Fundamentals of Physics Chapter 3 22 Uniform Circular Motion The velocity is changing, but the speed is not. Hence, the particle is accelerating. The acceleration is always directed radially inward. Centripetal Acceleration: The velocity is always tangent to the circular path. Let’s prove this.

23. 2009 Physics 2111 Fundamentals of Physics Chapter 3 23 Circular Motion

24. 2009 Physics 2111 Fundamentals of Physics Chapter 3 24 “Reference Frame B moves @ constant velocity wrt Reference Frame A” Relative Motion in One Dimension Differentiate wrt time

25. 2009 Physics 2111 Fundamentals of Physics Chapter 3 25 Relative Motion in 2 Dimensions Two observers watching the motion of particle P from the origins of reference frames A and B, while B moves at constant velocity v ba relative to A. Observers in different reference frames that move at constant velocity relative to each other will measure the same acceleration. Differentiate wrt time

26. 2009 Physics 2111 Fundamentals of Physics Chapter 3 26 Relative Motion v pw = 215 km/h,  south of east v wg = 65.0 km/hr 20 0 E of N What is v pg and 

27. 2009 Physics 2111 Fundamentals of Physics Chapter 3 27 Example y – y 0 = v 0 sin  0 t - ½ g t 2 R = x – x 0 = v 0 cos  0 t

28. 2009 Physics 2111 Fundamentals of Physics Chapter 3 28 Example: How fast should she be going?

29. 2009 Physics 2111 Fundamentals of Physics Chapter 3 29 Example: Can He Make the Jump? x – x 0 = v 0 cos  0 t y – y 0 = v 0 sin  0 t - ½ g t 2  0 = 0 v0v0 y0y0 y = 0 He can’t make it! t = 0.990 s x – x 0 = 4.5 m