2. Chapter 3 Motion in Two Dimensions (2D) Vectors and Projectile Motion x y

3. What is a …? Vector –Has Magnitude & Direction 12 m/s Northeast –Examples: Drawn as arrows Angles measured from 0  Scalar –No direction –Examples: 12 m/s tip tail 0° + 

4. Vector addition Vectors can be added together 2 ways: graphically or analytically (with math!) The answer to a vector addition problem is called the Resultant, which is also a vector, so it has a magnitude (size) and direction The Resultant (R) goes from the starting tail to the ending tip Any number of vectors can be added with these methods http://illuminations.nctm.org/Activity.aspx?id=6598 v1v1 v2v2 R

5. Graphical Addition Add using “Tip to Tail” method—tail of the next vector is drawn on the tip of the previous Draw to scale, keeping size and direction. Use ruler & protractor. Start at an origin! http://lectureonline.cl.msu.edu/~mmp/kap3/cd052a.htm Examples next

6. Vector Direction: Examples Write the vector angle in terms of 0 to 360° A. 50m North B. 100m West C.50m South D.100m East 0° Now add (sketch) A+D and B+C A B C D http://phet.colorado.edu/simulations/sims.php ?sim=Vector_Addition

7. Vector addition Properties of vectors: –Vectors can be moved parallel to themselves in a diagram –Vectors can be added in any order –To subtract a vector, add the vector pointing in the opposite direction (or negative) –Multiplying or dividing vectors by scalars results in vectors –http://www.nctm.org/standards/content.aspx?id=26779http://www.nctm.org/standards/content.aspx?id=26779 –http://illuminations.nctm.org/Activity.aspx?id=6598 stormhttp://illuminations.nctm.org/Activity.aspx?id=6598

8. Analytical Addition We’ll only look at vectors which are perpendicular to each other, so Right Triangles! Use Pythagorean theorem and/or Trigonometry to find exact values for Resultant SOH CAH TOA –One axis is x, the other is y—they are independent of each other –Make a sketch to help—add “tip to tail”, draw “ x ” first –Always indicate  from 0 , use sketch to reduce errors Carefully determine the correct quadrant for Resultant, R

9. Example: Perpendicular vectors Little red riding hood walks to grandma’s house 165 m North and 125 m to the East. What is her resultant displacement? Example in packet R = 207 m E N R  tan  =  y /  x = 165m / 125m  = tan -1 (1.32) = 52.9°

10. Next… When you know the vector, but want the x and y that make it up: Breaking the vectors into its pieces is called Resolving the vector Use Trig to get the horizontal and vertical parts, or components, of the original vector

11. How to add these? http://www.surendranath.org/Applets/Math/Vec torAddition/VA.html

12. Resolving Vectors into Components Vector A and its x and y components Often vector length and angle are known FORMULAS FOR  FROM X AXIS ONLY

13. Example What are the components that make up the given vector V? V = 10 m/sec 60° V x = V cos  = 10cos60 ° = 5 m/s V y = V sin  = 10sin60 ° = 8.66 m/s Examples cos  = V x / V sin  = V y / V

14. Adding Vectors at angles C:\Documents and Settings\anthoedw\My Documents\CP\temp-vector-addition-phet\vector-addition_en.html

15. Adding Vectors at angles Demonstration of the major steps The full procedure is on the next slide..\Vector Addition probl.ppt

16. Write all vector angles from 0  up to 360 degrees. 1.Resolve into x & y components for all vectors. Vx = V cos  and Vy = V sin  for  measured to x axis 2.Add all components for x, and all for y direction to find the totals in the x and y. 3.Calculate the length and angle of the resultant by using totals and the equation: Knowing total x and y, calculate Resultant length with Trig or Pythagorean Thm: By referring to the signs of the x and y totals of the resultant, determine the quadrant along with the appropriate angle from 0 to 360 degrees. EXAMPLES Method for Vector Addition

17. What is a Projectile?

18. Projectile motion Projectile : Moving object that is only affected by gravity. The motion of a projectile is determined only b y the object’s initial velocity and gravity ( g ). v i has v y and v x components. Use x & y dimensions to analyze motion. V ertical motion ( y ) of a projected object is independent of its horizontal motion. Vertical motion o f a projectile is nothing more than free fall (acceleration). Horizontal m otion ( x ) is constant v elocity. The common variable between the horizontal and vertical motion is time

19. Path of a Projectile The path of a projectile is called its trajectory. The trajectory of a projectile in free fall is a parabola. http://regentsprep.org/Regents/physics/phys06/amotproj/sld005.htm i Variables: ∆ x v x ∆ t ∆ y v iy v fy a y v i and  so also: v x = v i cos  v iy = v i sin 

20. Projectile Motion Problems The first type of projectile motion is: Horizontal Launch Projectiles –P–Path appears to be ½ a parabola –P–Projectile has no initial vertical velocity, v iy = 0 –V–Vertical is done like a dropped object (free fall) –P–Projectile has no horizontal acceleration, a x = 0 –H–Horizontal motion portion is done the same as a constant velocity object ( v ix = v fx = v x ) –U–Use previous equations with revised subscripts

21. Question ? If a gun is fired horizontally and simultaneously a bullet is dropped from the same height, which bullet hits the ground first?

22. Variables: ∆ x v x ∆ t ∆ y V iy = 0 v fy A y Angle  = 0

23. Projectile formulas x-direction y-direction ** Note: v i is the vector sum (resultant) of v x and v iy v f is the vector sum of v x and v fy Constant v Free fall

24. Example A ball is launched horizontally at 10 m/s off of a 1.2 m tall table. Calculate the following: A)The time in the air. B)How far from the table does the ball land (x)? C) What is the magnitude of the velocity of the ball just as it hits the ground? Examples in packet

25. Other horizontal examples Drop a bomb Shoot Throw or drive off a cliff….. All are initially going horizontally only, the x direction (an angle of 0  or 180  ) True Lies… two more questions…..\My Videos\robbiemadison motorcycle.mp4

26. Horizontal Projectile An airplane flying steadily and horizontally drops a bowling ball. If it is in free fall, where does it hit relative to the plane? In front, directly under, or behind?

27. Horizontal Projectile What would happen if you shoot a projectile from a moving vehicle and the projectile has the same speed as the vehicle, but opposite direction? https://www.youtube.com/watch?v=BLuI11 8nhzc https://www.youtube.com/watch?v=BLuI11 8nhzc

28. Projectile Motion Problems The second type of projectile motion is for objects Launched at an Angle (V i and  ) These objects will follow a full parabola The up ½ of the motion = the down ½ at corresponding heights, other than direction (it is symmetric) Initial velocity is a vector at an angle which must be resolved into x & y components Same equations work as for Horizontal projectiles, but v iy is not = 0. Use v x = v i ·cos , v iy = v i ·sin ..\My Videos\robbiemadison motorcycle.mp4

29.  y =0 Projectile Motion Problems http://jersey.uoregon.edu/newCa nnon/nc4.html http://jersey.uoregon.edu/newCa nnon/nc1.html http://www.mhhe.com/physsci/ph ysical/giambattista/proj/projectile. html Total time in the air ( t total ) = time to go up + time to go down v i = initial velocity (total), includes both v x and v iy v f = final velocity (total); = magnitude of v i (or v x + v fy )  y max = maximum height (top). Note: v y = 0  x max = total horizontal distance, called Range (R)

30. Range symmetry and maximum http://phet.colorad o.edu/simulations/ projectilemotion/p rojectile.swf

33. Problem Solving Angled launch problems are solved similar to horizontal Same procedure except for the start: Resolve velocity vector into x & y components ( v iy is not zero) Use symmetry and V y = 0 at top when needed examples

34. Example A cannon ball gets fired at a 30 degree angle at 26 m/sec. A)How far away does it land? B)How long is it in the air? C)What is the maximum height? D)How fast is it moving when it hits the ground? E)How fast is the ball moving and how high is the ball 0.6 sec after launch? http://jersey.uoregon.edu/newCa nnon/nc4.html http://jersey.uoregon.edu/newCa nnon/nc1.html

35. More practice: Practice Problem 1: A golf ball is hit with a velocity of 31 m/s at 48  above the horizontal. –(A) Find how long it is in the air. –(B) How high did the ball go? –(C) What was the range of the ball? Challenge Problem: A cannon shoots a pumpkin at an angle of 30 degrees which lands 120 m from the cannon. –A) With what velocity was the projectile fired? –B) What is the pumpkin’s velocity as it strikes the ground? –https://www.youtube.com/watch?v=b0xmTzSXL0Ehttps://www.youtube.com/watch?v=b0xmTzSXL0E

36. Monkey Hunter!! http://www.phy.ntnu.edu.tw/ntnujava/index.php?PHPSESSID=04e5251cd5a765c0ee22dd61 a2040312&topic=144.msg721#msg721 http://www.physics.umn.edu/outreach/pforce/circus/ http://www.mhhe.com/physsci/physical/giambattista/pr oj/projectile.html http://www.waowen.screamin g.net/revision/force&motion/m andh.htm