2. Combining Motion and Vector concepts 2D Motion

3. Moving Motion Forward Velocity, Displacement and Acceleration are VECTORS Vectors have magnitude AND direction So far, direction has been positive and negative

4. Motion with Vectors Motion concepts combine with Vector concepts to allow us to represent motion in multiple directions simultaneously Two (2) major concepts will be addressed: – 1. Moving in two directions at once (e.g. projectiles) – 2. Moving with respect to (wrt) a non-stationary object (“relative” motion)

5. How High – How Far 2 nd significant project of semester – Launch an object – Measure distance and time – Calculate everything else – Take video or pictures – Upload to Gaggle or Edmodo (zdm2ni) Two daily grades (undroppable) Due Tuesday, October 29 th

6. What is Projectile Motion?

7. Projectile Motion Two-dimensional motion of an object – Vertical – Horizontal

11. Projectiles Airborne objects that move as a result of their own inertia and gravity. They exhibit two dimensional motion. – Horizontal and vertical motion Horizontal and vertical motion are independent of one another. – The resultant motion is a combination of horizontal and vertical motion.

12. Components of Projectile Motion Projectiles have: – vertical velocities that increase and decrease because they are influenced by gravitational forces. – horizontal velocities remain constant because they are not influenced by gravitational forces.

13. Types of Projectile Motion Horizontal – Motion of a ball rolling freely along a level surface – Horizontal velocity is ALWAYS constant Vertical – Motion of a freely falling object – Force due to gravity – Vertical component of velocity changes with time Parabolic – Path traced by an object accelerating only in the vertical direction while moving at constant horizontal velocity

14. Displacement for Projectiles

15. Examples of Projectile Motion Launching a Cannon ball

16. Simulators and examples Music analogy http://www.geogebra.org/en/upload/files/nike nuke/projectile06d.html http://www.geogebra.org/en/upload/files/nike nuke/projectile06d.html http://www.cs.utah.edu/~zachary/isp/applets/ Cannon/Cannon.html http://www.cs.utah.edu/~zachary/isp/applets/ Cannon/Cannon.html

17. Velocity “Splits” Have a fixed starting velocity Splits between X and Y initial velocities using “Triangle” rules

18. Horizontal (X) motion X-distance is called “Range” Moves at constant velocity/speed/rate – i.e. ZERO acceleration V TOTAL VXVX VYVY θ

19. Vertical (Y) Motion Y-direction called “Height” Y velocity changes (accelerates) because of Gravity (g) = - 9.8 m/s 2 ≈ -10 m/s 2 Y velocity (speed) “stops” (=0) at “top” of trajectory Returns to same “magnitude” but different sign when returns to “launch level” V TOTAL VXVX VYVY θ

20. Horizontally Launched Projectiles Rolls off table/cliff Thrown/Fired Horizontally – Launch angle = 0º ALL initial velocity is in X direction – V y-initial = 0

21. Horizontally Launched Projectiles Equations Simplify

22. Horizontally Launched Projectiles Sample Problem A dart gun is fired from a 1.9 meter table and lands 2.8 meters from the table. – What is the “muzzle velocity” of the dart gun? – How far would the dart travel if fired from 2.3 m?

23. Horizontally Launched Projectiles Sample Problem A dart gun is fired from a 1.9 meter table and lands 2.8 meters from the table. – What is the “muzzle velocity” of the dart gun? – How far would the dart travel if fired from 2.3 m?

24. ADDITIONAL “OFFICIAL” FORMULAS

25. Equations – X and Y components X- Component Y- Component Note: acceleration due to gravity is

26. Equations – Combined (vector)

27. Equations X- Component Y- Component Vectors Note: acceleration due to gravity is

28. Strategies for Applying Kinematic Equations in 2 Dimensions Make a drawing of the situation. Decide which directions to be called positive and negative. Do this for both the x and y directions. Remember that the time variable is the same for both x and y. Make note of all the assumed information. – Starts at rest. – Stops When motion is divided into two segments, remember the final velocity for one segment becomes the initial velocity for the next segment. Keep in mind there might be two possible answers form the same problem. Choose the answer that matches the physical situation at hand.

29. Factors Affecting Projectile Motion What two factors would affect projectile motion? – Angle – Initial velocity Initial Velocity Angle

30. Launch Angle Launch Angle to maximize distance is 45° Complementary angles (add up to 90°) will launch to the same horizontal distance

31. Class Exercise An object is fired from the ground at 100 meters per second at an angle of 30 degrees with the horizontal Calculate the horizontal and vertical components of the initial velocity After 2.0 seconds, how far has the object traveled in the horizontal direction? How high is the object at this point?

32. Applications Any Ideas?

33. Solution Part a Part b Part c