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# Vectors and projectile motion and worked examples

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### Vectors and projectile motion and worked examples

1. 1. Stage 2 Physics Section 1Motion in Two Dimensions
2. 2. What is motion?What types of motion are there?What causes motion?How to we describe motion in Physics?
3. 3. What is a vector?Vector quantities have magnitude (size) anddirection.Scalar quantities have magnitude only.Length represents the vectors magnitude.
4. 4. Scalar VectorDistance DisplacementSpeed VelocityMass Acceleration ForceTime weight
5. 5. Velocity vector of an angry bird
6. 6. Resultant of Two VectorsThe resultant is the sum or the combined addition of two vectorquantitiesVectors in the same direction: 6N 4N = 10 N Vectors in opposite directions: 6m 10 m =
7. 7. Vectors• Vectors can be oriented to the gravitational field (up, down or some angle to the horizontal) or compass points (NESW). 5 ms-1 5 ms-1 30o above the horizontal
8. 8. Velocity Vectors• Velocity can be resolved into its horizontal and vertical components at any instant. v vV vH
9. 9. SOHCAHTOA Hypotenuse Opposite Adjacent
10. 10. v vV = v sinvH = v cos
11. 11. Example 1Resolve the following velocity vector into its horizontal and vertical components 30o
12. 12. Example 1Resolve the following velocity vector into its horizontal and vertical components 30o
13. 13. This problem can be solved in two ways(and you need to be able to do both)1. Scale Diagram2. Trigonometry
14. 14. 1. Scale diagramBy drawing a vector diagram (using aprotractor and a ruler) to scale we cansimply measure the size of thecomponents ideally the vector should be10 cm or larger (for accuracy)
15. 15. 2. Trigonometry vvertical = v sin = 40 sin 30o = 20 m s-1 vhorizontal = v cos30o = 40 cos 30o = 34.6 m s-1
16. 16. Example 2Determine the velocity vector with initialhorizontal velocity component of 50 ms –1and vertical 20 ms-1.
17. 17. v=? vv= 20 m s-1 vh= 50 m s- 1
18. 18. 1. Scale diagram Again we could accurately draw the figure and measure the resultant length and angle to find the direction of v.(Note: You need to have a clear Perspex ruler and a protractor for EVERY test and exam)
19. 19. 2. Pythagoras & Trigonometry By Pythagoras theory:v2 = vV2 + vH2 v = v V 2 + vH 2 = 202 + 502 = 2900 = 53.9 m s-1 v=? vv= 20 m s-1 vh= 50 m s- 1
20. 20. tan vV/vHtan = 20/50 = 21.8oie. v = 53.9 m s-1 at 21.8o above the horizontal
21. 21. Summary – Vectors in 2D• Given any vector quantity in 2D it can be resolved into horizontal and vertical components eg displacement, force, fields etc• Given the horizontal & vertical components you can determine magnitude and direction of the vector (formula)
22. 22. Motion in a UniformGravitational Field
23. 23. In the absence of gravity objects move with constant velocity in a straight line. An object will remain at rest, or continue to move at a constant velocity, unless a net force acts on it.Note: The following is all in the absence of air resistance.
24. 24. When an object falls under theinfluence of gravity, the verticalforce causes a constantacceleration
25. 25. vH The resultant motion is a combination of both horizontal and vH verticalvV vV components
26. 26. Horizontal Projection While the vertical component undergoes constantIf an object is acceleration.projectedhorizontally, thehorizontal componentmoves with constantvelocity.
27. 27. Three equations of motionNote that all the equations have “a” in them –they only apply under CONSTANT acceleration
28. 28. Constant vertical acceleration Vertical formulae 2 v 2 v0 2as 1 2Horizontal velocity is constant s vt at 2Horizontal formula v vo at s vH t
29. 29. Three equations of motion
30. 30. Learning Symbols in PhysicsQuantity Quantity Symbol Units Unit symbol
31. 31. ExampleA stone is dropped down a well and takes 3seconds to hit the ground.a) How fast does it hit the bottom?b) How deep is the well?
32. 32. A stone is dropped down a well attakes 3 seconds to hit the ground.a) How fast does it hit the bottom?
33. 33. b) How deep is the well?
34. 34. An arrow is fired upwards at 50ms-1.a) How high does the arrow fly?
35. 35. An arrow is fired upwards at 50ms-1.b) How long does the arrow take to hit theground?
36. 36. Projectile Motion ProblemsExcept for time, everything can be separated into horizontal and vertical components and treated separately. sV = height V0 s H = range t = time of flight
37. 37. Projectile Motion ProblemsHorizontal projection: down is +veUni-level projection: Up is considered positive, and down is negative.(Acceleration due to gravity aV = -9.8ms-2) sV = height V0 s H = range t = time of flight
38. 38. Projectile Motion ProblemsAt the top of the parabolic path, vV= 0 ms-1 1 vV 0ms 2 aV 9.8ms sV height V0 sH range t = time of flight
39. 39. Projectile Motion ProblemsRemember the time of flight is the time it takes to go up+ down. 1 vV 0ms 2 aV 9.8ms sV height V0 sH range t = time of flight
40. 40. Example 1
41. 41. Bi-level projection• An object is projected at a height
42. 42. Maximum RangeTo get the maximum range sH max in a vacuum(no air resistance) the launch angle must be 45o sH max
43. 43. For a projectile launched at ground level find by sample calculation the launch angle that results in a maximum range
44. 44. Pairs of launch angles that yield the same range add up to 90o α + θ = ranges Projectile 90o for various angles of launch 500 450 400 350 300height 250 200 150 100 50 0 0 200 400 600 800 1000 1200 range
45. 45. α + θ = 90oFind the launch angle that yields the same range as 32oθ = 32 α=? α + θ = 90o
46. 46. The Effect of Air ResistanceAir resistance acts in the opposite direction to motion. vertical horizontal
47. 47. The Effect of Air Resistance vertical horizontalThis decreases the• height• rangeSlight decrease in time of flight of the projectile
48. 48. The magnitude of Fair resistance Fair resistance
49. 49. Speed
50. 50. ShapeAerodynamicteardrop
51. 51. SizeMore surface area = more air resistance
52. 52. TextureSmoothRough
53. 53. Air densityLow air density = less air resistanceHigh air density = high air resistance
54. 54. Projectiles in SportConsider the effect of launch height on range
55. 55. As the object has further to fall tflight isincreased.As the object is in the air for longer ittravels farther.
56. 56. 45o 41o For objects at h=0 the optimal angle is 45oFor heights › 0θ max height is less than 45o