2. Pointing the Way Vectors

3. Representing Vectors Vectors are represented on paper by arrows – Direction = WAY THE ARROW POINTS – Magnitude = ARROW LENGTH Examples of Vectors – Displacement – Velocity – Acceleration

4. Vector Components Break vector into two perpendicular components using X-Y system –A–A X = A cos θ –A–A Y = A sin θ Example –v–v = 6.0 m/s at 30° v x = 6.0 m/s (cos 30°) = 5.2 m/s v y = 6.0 m/s (sin 30°) = 3.0 m/s 30° 6.0 m/s vXvX vYvY

5. Vectors can be added together by adding their COMPONENTS Results are used to find – RESULTANT MAGNITUDE – RESULTANT DIRECTION Adding Vectors Using Components

6. Checking Your References Relative Motion

7. How would Homer know that he is hurtling through interstellar space if his speed were constant? Without a window, he wouldn’t! All of the Laws of Motion apply within his FRAME of REFERENCE

8. Do you feel like you are motionless right now? ALL Motion is RELATIVE ! The only way to define motion is by changing position… The question is changing position relative to WHAT?!? You are moving at about 1000 miles per hour relative to the center of the Earth! The Earth is hurtling around the Sun at over 66,000 miles per hour! MORE MOTION!!!

9. Example #1 A train is moving east at 25 meters per second. A man on the train gets up and walks toward the front at 2 meters per second. What is his velocity? – Depends on what we want to relate his speed to!!! +2 m/s (relative to a fixed point on the train) +27 m/s (relative to a fixed point on the Earth) v train = +25 m/sv person = +2 m/s

10. Example #2 A passenger on a 747 that is traveling east at 230 meters per second walks toward the lavatory at the rear of the airplane at 1.5 meters per second. What is the passenger’s velocity? – Again, depends on how you look at it! -1.5 m/s (relative to a fixed point in the 747) +228.5 m/s (relative to a fixed point on the Earth)

11. Non-Parallel Vectors What happens to the aircraft’s forward speed when the wind changes direction? v thrust No wind – plane moves with velocity that comes from engines v wind Wind in same direction as plane – adds to overall velocity! Wind is still giving the plane extra speed, but is also pushing it SOUTH. Wind is now NOT having any effect on forward movement, but pushes plane SOUTH. Wind is now slowing the plane somewhat AND pushing it SOUTH. Wind is now working against the aircraft thrust, slowing it down, but causing no drift.

12. Perpendicular Kinematics Critical variable in multi dimensional problems is TIME. We must consider each dimension SEPARATELY, using TIME as the only crossover VARIABLE.

13. Example A swimmer moving at 0.5 meters per second swims across a 200 meter wide river. 200 m v s = 0.5 m/s How long will it take the swimmer to get across? t =0 The time to cross is unaffected! The swimmer still arrives on the other bank in 400 seconds. What IS different? Now, assume that as the swimmer moves ACROSS the river, a current pushes him DOWNSTREAM at 0.1 meter per second. v c = 0.1 m/s The arrival POINT will be shifted DOWNSTREAM!

14. Over the Edge Horizontal Projectiles

15. A red ball rolls off the edge of a table What does its path look like as it falls? Parabolic path

16. As the red ball rolls off the edge, a green ball is dropped from rest from the same height at the same time Which one will hit the ground first? They will hit at the SAME TIME!!!

17. The same time?!? How?!? v ix The red ball has an initial HORIZONTAL velocity (v ix ) But does not have any initial VERTICAL velocity (v iy = 0) The green ball falls from rest and has no initial velocity IN EITHER DIRECTION! v iy and v ix = 0

18. One Dimension at a Time Both balls begin with no VERTICAL VELOCITY Both fall the same VERTICAL DISTANCE This means that the time of flight IS THE SAME IN BOTH CASES

19. Example #1 (not in notes) A bullet is fired horizontally from a gun that is 1.7 meters above the ground with a velocity of 55 meters per second. At the same time that the bullet is fired, the shooter drops an identical bullet from the same height. Prove that the two bullets hit the ground at the same time. For both bullets… v i = 0 a = -9.81 m/s 2 d = -1.7 m

20. An airplane making a supply drop to troops behind enemy lines is flying with a speed of 300 meters per second at an altitude of 3.0 kilometers. – How far from the drop zone should the aircraft drop the supplies? Need time from vertical d = v i t + ½ at 2 3000 m = 0 + ½ (-9.81 m/s 2 )t 2 t = 24.7 s Use time in horizontal d = v i t + ½ at 2 d = (300 m/s)(24.7 s) + 0 d = 7410 m Example #2

21. Example #3 A stuntman jumps off the edge of a 45 meter tall building to an air mattress that has been placed on the street below at 15 meters from the edge of the building. – What minimum initial velocity does he need in order to make it onto the air mattress? Need time from vertical d = v i t + ½ at 2 45 m = 0 + ½ (-9.81 m/s 2 )t 2 t = 3.03 s Use time to find v v = d / t v = 15 m / 3.03 s v = 4.95 m /s

22. Example #4 An astronaut on the Moon throws a wrench horizontally with a speed of 0.5 meters per second from a height of 1.5 meters. The astronaut simultaneously drops a feather from the same height. Which object will hit first? Why? With no air resistance and identical vertical variables they hit the ground at the same time.

23. Fire Away!!! Ground Launched Projectiles

24. What is a projectile? A flying object with no means of propulsion after its launch. Can be launched with any VELOCITY and at ANY ANGLE May be subjected to AIR RESISTANCE

25. Initial velocity 30° What is the horizontal part of the soccer ball’s initial velocity? v ix = v i cos θ What is the vertical part of the soccer ball’s initial velocity? v iy = v i sin θ 12 m/s 6 m/s 10.4 m/s

26. What assumptions can we make? Horizontal Initial velocity = v i cos θ Acceleration = 0 Vertical Initial velocity = v i sin θ Acceleration = -9.81 m/s 2 Vertical speed = 0 at top

27. Projectile Vector Diagram t 1/2 v fy = 0 (at top) x y a = 0 a = -9.81 m/s 2 v i = v i cos θ v i = v i sin θ v f = 0 (at top) t tot = 2t 1/2 t tot