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# 12 x1 t07 01 projectile motion (2012)

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### 12 x1 t07 01 projectile motion (2012)

1. 1. Projectile Motion
2. 2. Projectile Motion y x
3. 3. Projectile Motion y  x
4. 4. Projectile Motion y  x
5. 5. Projectile Motion y maximum range   45  xInitial conditions
6. 6. Projectile Motion y maximum range   45  xInitial conditions when t = 0 v 
7. 7. Projectile Motion y maximum range   45  xInitial conditions when t = 0 v  y   x
8. 8. Projectile Motion y maximum range   45  xInitial conditions when t = 0  x v  cos  y v  x  v cos   x
9. 9. Projectile Motion y maximum range   45  xInitial conditions when t = 0  x  y  cos  sin  v  y v v  x  v cos  y  v sin    x
10. 10. Projectile Motion y maximum range   45  xInitial conditions when t = 0  x  y  cos  sin  v  y v v  x  v cos  y  v sin    x x0
11. 11. Projectile Motion y maximum range   45  xInitial conditions when t = 0  x  y v  cos  sin   y v v  x  v cos  y  v sin    x x0 y0
12. 12.   0x    g y
13. 13.   0x    g yx  c1
14. 14.   0x    g yx  c1 y   gt  c2 
15. 15.   0 x    g y x  c1  y   gt  c2 when t  0, x  v cos  y  v sin  
16. 16.   0 x    g y x  c1  y   gt  c2 when t  0, x  v cos  y  v sin   c1  v cos x  v cos 
17. 17.   0 x    g y x  c1  y   gt  c2 when t  0, x  v cos  y  v sin   c1  v cos c2  v sin  x  v cos  y   gt  v sin  
18. 18.   0 x    g y x  c1  y   gt  c2 when t  0, x  v cos  y  v sin   c1  v cos c2  v sin  x  v cos  y   gt  v sin   1 2 x  vt cos  c3 y   gt  vt sin   c4 2
19. 19.   0 x    g y x  c1  y   gt  c2 when t  0, x  v cos  y  v sin   c1  v cos c2  v sin  x  v cos  y   gt  v sin   1 2 x  vt cos  c3 y   gt  vt sin   c4 2 when t  0, x  0 y0
20. 20.   0 x    g y x  c1  y   gt  c2 when t  0, x  v cos  y  v sin   c1  v cos c2  v sin  x  v cos  y   gt  v sin   1 2 x  vt cos  c3 y   gt  vt sin   c4 2 when t  0, x  0 y0 c3  0 x  vt cos
21. 21.   0 x    g y x  c1  y   gt  c2 when t  0, x  v cos  y  v sin   c1  v cos c2  v sin  x  v cos  y   gt  v sin   1 2 x  vt cos  c3 y   gt  vt sin   c4 2 when t  0, x  0 y0 c3  0 c4  0 x  vt cos 1 y   gt 2  vt sin  2
22. 22.   0 x    g y x  c1  y   gt  c2 when t  0, x  v cos  y  v sin   c1  v cos c2  v sin  x  v cos  y   gt  v sin   1 x  vt cos  c3 y   gt 2  vt sin   c4 2 when t  0, x  0 y0 c3  0 c4  0 x  vt cos 1 2 y   gt  vt sin  2 Note: parametric coordinates of a parabola
23. 23.   0 x    g y x  c1  y   gt  c2  when t  0, x  v cos  y  v sin   c1  v cos c2  v sin  x  v cos  y   gt  v sin   1 x  vt cos  c3 y   gt 2  vt sin   c4 2 when t  0, x  0 y0 c3  0 c4  0 x  vt cos 1 2 y   gt  vt sin  2 Note: parametric coordinates of a parabola xt v cos 
24. 24.   0 x    g y x  c1  y   gt  c2  when t  0, x  v cos  y  v sin   c1  v cos c2  v sin  x  v cos  y   gt  v sin   1 2 x  vt cos  c3 y   gt  vt sin   c4 2 when t  0, x  0 y0 c3  0 c4  0 x  vt cos 1 2 y   gt  vt sin  2 Note: parametric coordinates of a parabola x gx 2 x sin t y 2  v cos  2v cos  cos  2 gx 2 y   2 sec 2   x tan  2v
25. 25.   0 x    g y x  c1  y   gt  c2  when t  0, x  v cos  y  v sin   c1  v cos c2  v sin  x  v cos  y   gt  v sin   1 x  vt cos  c3 y   gt 2  vt sin   c4 2 when t  0, x  0 y0 c3  0 c4  0 x  vt cos 1 2 y   gt  vt sin  2 Note: parametric coordinates of a parabola x gx 2 x sin t y 2  gx 2 v cos  2v cos  cos  y   2  tan 2   1  x tan  2 2v gx 2 y   2 sec 2   x tan  2v
26. 26. Common Questions
27. 27. Common Questions(1) When does the particle hit the ground?
28. 28. Common Questions(1) When does the particle hit the ground? Particle hits the ground when y  0
29. 29. Common Questions(1) When does the particle hit the ground? Particle hits the ground when y  0(2) What is the range of the particle?
30. 30. Common Questions(1) When does the particle hit the ground? Particle hits the ground when y  0(2) What is the range of the particle? i  find when y  0
31. 31. Common Questions(1) When does the particle hit the ground? Particle hits the ground when y  0(2) What is the range of the particle? i  find when y  0 ii  substitute into x
32. 32. Common Questions(1) When does the particle hit the ground? Particle hits the ground when y  0(2) What is the range of the particle? i  find when y  0 roots of the quadratic ii  substitute into x
33. 33. Common Questions(1) When does the particle hit the ground? Particle hits the ground when y  0(2) What is the range of the particle? i  find when y  0 ii  substitute into x(3) What is the greatest height of the particle?
34. 34. Common Questions(1) When does the particle hit the ground? Particle hits the ground when y  0(2) What is the range of the particle? i  find when y  0 ii  substitute into x(3) What is the greatest height of the particle? i  find when y  0 
35. 35. Common Questions(1) When does the particle hit the ground? Particle hits the ground when y  0(2) What is the range of the particle? i  find when y  0 ii  substitute into x(3) What is the greatest height of the particle? i  find when y  0  ii  substitute into y
36. 36. Common Questions(1) When does the particle hit the ground? Particle hits the ground when y  0(2) What is the range of the particle? i  find when y  0 roots of the quadratic ii  substitute into x(3) What is the greatest height of the particle? i  find when y  0  vertex of the parabola ii  substitute into y
37. 37. Common Questions(1) When does the particle hit the ground? Particle hits the ground when y  0(2) What is the range of the particle? i  find when y  0 ii  substitute into x(3) What is the greatest height of the particle? i  find when y  0  ii  substitute into y(4) What angle does the particle make with the ground?
38. 38. Common Questions(1) When does the particle hit the ground? Particle hits the ground when y  0(2) What is the range of the particle? i  find when y  0 ii  substitute into x(3) What is the greatest height of the particle? i  find when y  0  ii  substitute into y(4) What angle does the particle make with the ground? i  find when y  0
39. 39. Common Questions(1) When does the particle hit the ground? Particle hits the ground when y  0(2) What is the range of the particle? i  find when y  0 ii  substitute into x(3) What is the greatest height of the particle? i  find when y  0  ii  substitute into y(4) What angle does the particle make with the ground? i  find when y  0 ii  substitute into y 
40. 40. Common Questions(1) When does the particle hit the ground? Particle hits the ground when y  0(2) What is the range of the particle? i  find when y  0 ii  substitute into x(3) What is the greatest height of the particle? i  find when y  0  ii  substitute into y(4) What angle does the particle make with the ground? i  find when y  0 ii  substitute into y   y iii  tan   y  x  x
41. 41. Common Questions(1) When does the particle hit the ground? Particle hits the ground when y  0(2) What is the range of the particle? i  find when y  0 roots of the quadratic ii  substitute into x(3) What is the greatest height of the particle? i  find when y  0  vertex of the parabola ii  substitute into y(4) What angle does the particle make with the ground? i  find when y  0 (i) find slope of the tangent ii  substitute into y   y  ii  m  tan iii  tan   y  x  x
42. 42. Summary
43. 43. Summary A particle undergoing projectile motion obeys
44. 44. Summary A particle undergoing projectile motion obeys   0 x and    g y
45. 45. Summary A particle undergoing projectile motion obeys   0 x and    g y with initial conditions
46. 46. Summary A particle undergoing projectile motion obeys   0 x and    g y with initial conditions x  v cos  and y  v sin  
47. 47. Summary A particle undergoing projectile motion obeys   0 x and    g y with initial conditions x  v cos  and y  v sin  e.g. A ball is thrown with an initial velocity of 25 m/s at an angle of 3   tan 1 to the ground. Determine;. 4
48. 48. Summary A particle undergoing projectile motion obeys   0 x and    g y with initial conditions x  v cos  and y  v sin   e.g. A ball is thrown with an initial velocity of 25 m/s at an angle of 3   tan 1 to the ground. Determine;. 4a) greatest height obtained
49. 49. Summary A particle undergoing projectile motion obeys   0 x and    g y with initial conditions x  v cos  and y  v sin   e.g. A ball is thrown with an initial velocity of 25 m/s at an angle of 3   tan 1 to the ground. Determine;. 4 a) greatest height obtainedInitial conditions
50. 50. Summary A particle undergoing projectile motion obeys   0 x and    g y with initial conditions x  v cos  and y  v sin   e.g. A ball is thrown with an initial velocity of 25 m/s at an angle of 3   tan 1 to the ground. Determine;. 4 a) greatest height obtainedInitial conditions 5 3 3   tan 1 4 4
51. 51. Summary A particle undergoing projectile motion obeys   0 x and    g y with initial conditions x  v cos  and y  v sin   e.g. A ball is thrown with an initial velocity of 25 m/s at an angle of 3   tan 1 to the ground. Determine;. 4 a) greatest height obtainedInitial conditions x  v cos  5 3 3   tan 1 4 4
52. 52. Summary A particle undergoing projectile motion obeys   0 x and    g y with initial conditions x  v cos  and y  v sin   e.g. A ball is thrown with an initial velocity of 25 m/s at an angle of 3   tan 1 to the ground. Determine;. 4 a) greatest height obtainedInitial conditions x  v cos  5 3 x  25  4     tan 1 3 5 4  20m/s 4
53. 53. Summary A particle undergoing projectile motion obeys   0 x and    g y with initial conditions x  v cos  and y  v sin   e.g. A ball is thrown with an initial velocity of 25 m/s at an angle of 3   tan 1 to the ground. Determine;. 4 a) greatest height obtainedInitial conditions x  v cos  y  v sin   5 3 x  25  4     tan 1 3 5 4  20m/s 4
54. 54. Summary A particle undergoing projectile motion obeys   0 x and    g y with initial conditions x  v cos  and y  v sin   e.g. A ball is thrown with an initial velocity of 25 m/s at an angle of 3   tan 1 to the ground. Determine;. 4 a) greatest height obtainedInitial conditions x  v cos  y  v sin   5 3 x  25  4 y  25  3       tan 1 3 5 5 4  20m/s  15m/s 4
55. 55.   0x   10 y
56. 56.   0x   10 yx  c1
57. 57.   0x   10 yx  c1 y  10t  c2 
58. 58.   0 x   10 y x  c1  y  10t  c2 when t  0, x  20  y  15 
59. 59.   0 x   10 y x  c1  y  10t  c2 when t  0, x  20  y  15  c1  20 x  20 
60. 60.   0 x   10 y x  c1  y  10t  c2 when t  0, x  20  y  15  c1  20 c2  15 x  20  y  10t  15 
61. 61.   0 x   10 y x  c1  y  10t  c2 when t  0, x  20  y  15  c1  20 c2  15 x  20  y  10t  15  x  20t  c3
62. 62.   0 x   10 y x  c1  y  10t  c2 when t  0, x  20  y  15  c1  20 c2  15 x  20  y  10t  15  x  20t  c3 y  5t 2  15t  c4
63. 63.   0 x   10 y x  c1  y  10t  c2 when t  0, x  20  y  15  c1  20 c2  15 x  20  y  10t  15  x  20t  c3 y  5t 2  15t  c4 when t  0, x  0 y0
64. 64.   0 x   10 y x  c1  y  10t  c2 when t  0, x  20  y  15  c1  20 c2  15 x  20  y  10t  15  x  20t  c3 y  5t 2  15t  c4 when t  0, x  0 y0 c3  0 x  20t
65. 65.   0 x   10 y x  c1  y  10t  c2 when t  0, x  20  y  15  c1  20 c2  15 x  20  y  10t  15  x  20t  c3 y  5t 2  15t  c4 when t  0, x  0 y0 c3  0 c4  0 x  20t y  5t 2  15t
66. 66.   0 x   10 y x  c1  y  10t  c2  when t  0, x  20  y  15  c1  20 c2  15 x  20  y  10t  15  x  20t  c3 y  5t 2  15t  c4 when t  0, x  0 y0 c3  0 c4  0 x  20t y  5t 2  15tgreatest height occurs when y  0 
67. 67.   0 x   10 y x  c1  y  10t  c2  when t  0, x  20  y  15  c1  20 c2  15 x  20  y  10t  15  x  20t  c3 y  5t 2  15t  c4 when t  0, x  0 y0 c3  0 c4  0 x  20t y  5t 2  15tgreatest height occurs when y  0   10t  15  0 3 t 2
68. 68.   0 x   10 y x  c1  y  10t  c2  when t  0, x  20  y  15  c1  20 c2  15 x  20  y  10t  15  x  20t  c3 y  5t 2  15t  c4 when t  0, x  0 y0 c3  0 c4  0 x  20t y  5t 2  15tgreatest height occurs when y  0  3 3 2  3  10t  15  0 when t  , y  5   15  2 2  2 3 45 t  2 4
69. 69.   0 x   10 y x  c1  y  10t  c2  when t  0, x  20  y  15  c1  20 c2  15 x  20  y  10t  15  x  20t  c3 y  5t 2  15t  c4 when t  0, x  0 y0 c3  0 c4  0 x  20t y  5t 2  15tgreatest height occurs when y  0  3  3 2  3  10t  15  0 when t  , y  5   15  2 2  2 3 45 t  2 4 1  greatest height is 11 m above the ground 4
70. 70. b) range
71. 71. b) range time of flight is 3 seconds
72. 72. b) range time of flight is 3 seconds when t  3, x  203  60
73. 73. b) range time of flight is 3 seconds when t  3, x  203  60  range is 60m
74. 74. b) range time of flight is 3 seconds when t  3, x  203  60  range is 60m 1c) velocity and direction of the ball after second 2
75. 75. b) range time of flight is 3 seconds when t  3, x  203  60  range is 60m 1c) velocity and direction of the ball after second 2 1 1 when t  , x  20  y  10   15  2  2  10
76. 76. b) range time of flight is 3 seconds when t  3, x  203  60  range is 60m 1c) velocity and direction of the ball after second 2 1 1 when t  , x  20  y  10   15  10 5 2  2 10  10  20
77. 77. b) range time of flight is 3 seconds when t  3, x  203  60  range is 60m 1c) velocity and direction of the ball after second 2 1 1 when t  , x  20  y  10   15  10 5 2  2 10  10  1 tan   20 2   2634
78. 78. b) range time of flight is 3 seconds when t  3, x  203  60  range is 60m 1c) velocity and direction of the ball after second 2 1 1 when t  , x  20  y  10   15  10 5 2  2 10  10  1 tan   20 2   2634 1  after second, velocity  10 5m/s and it is traveling at 2 an angle of 2634 to the horizontal
79. 79. d) cartesian equation of the path
80. 80. d) cartesian equation of the path x  20t x t 20
81. 81. d) cartesian equation of the path x  20t y  5t 2  15t 2 x  x   15 x  t y  5    20  20   20   x 2 3x y  80 4
82. 82. d) cartesian equation of the path x  20t y  5t 2  15t 2 x  x   15 x  t y  5    20  20   20   x 2 3x y  80 4
83. 83. d) cartesian equation of the path x  20t y  5t 2  15t 2 x  x   15 x  t y  5    20  20   20   x 2 3x y  80 4
84. 84. d) cartesian equation of the path x  20t y  5t 2  15t 2 x  x   15 x  t y  5    20  20   20   x 2 3x y  80 4Using the cartesian equation to solve the problem
85. 85. d) cartesian equation of the path x  20t y  5t 2  15t 2 x  x   15 x  t y  5    20  20   20   x 2 3x y  80 4Using the cartesian equation to solve the problema) greatest height is y value of the vertex
86. 86. d) cartesian equation of the path x  20t y  5t 2  15t 2 x  x   15 x  t y  5    20  20   20   x 2 3x y  80 4Using the cartesian equation to solve the problema) greatest height is y value of the vertex 2 3 1     4( )(0) 4 80 9  16
87. 87. d) cartesian equation of the path x  20t y  5t 2  15t 2 x  x   15 x  t y  5    20  20   20   x 2 3x y  80 4Using the cartesian equation to solve the problema) greatest height is y value of the vertex  2 y 3 1 4a     4( )(0) 4 80 9 20    9 16 1  16 45  4
88. 88. d) cartesian equation of the path x  20t y  5t 2  15t 2 x  x  x t y  5   15  20  20   20   x 2 3x y  80 4Using the cartesian equation to solve the problema) greatest height is y value of the vertex  2 y 3 1 4a     4( )(0) 4 80 9 20    9 16 1  16 45  4 1  greatest height is 11 m above the ground 4
89. 89. b) range
90. 90. b) range  x 2 3xy  80 4 x x   3  4 20 
91. 91. b) range  x 2 3xy  80 4 x x   3  4 20 roots are 0 and 60 range is 60m
92. 92. 1b) range c) velocity and direction of the ball after second 2  x 2 3xy  80 4 x x   3  4 20 roots are 0 and 60 range is 60m
93. 93. 1b) range c) velocity and direction of the ball after second 2  x 2 3x 1 when t  , x  10y  80 4 2 x x   3  4 20 roots are 0 and 60 range is 60m
94. 94. 1b) range c) velocity and direction of the ball after second 2  x 2 3x 1 when t  , x  10  x 2 3xy  y  80 4 2 80 4 x x  dy  x 3  3    4 20  dx 40 4roots are 0 and 60 range is 60m
95. 95. 1b) range c) velocity and direction of the ball after second 2  x 2 3x 1 when t  , x  10  x 2 3xy  y  80 4 2 80 4 x x  dy  x 3  3    4 20  dx 40 4roots are 0 and 60 dy 1 when x  10,  range is 60m dx 2
96. 96. 1b) range c) velocity and direction of the ball after second 2  x 2 3x 1 when t  , x  10  x 2 3xy  y  80 4 2 80 4 x x  dy  x 3  3    4 20  dx 40 4 dy 1 1roots are 0 and 60 when x  10,  tan   range is 60m dx 2 2   2634
97. 97. 1b) range c) velocity and direction of the ball after second 2  x 2 3x 1 when t  , x  10  x 2 3xy  y  80 4 2 80 4 x x  dy  x 3  3    4 20  dx 40 4 dy 1 1roots are 0 and 60 when x  10,  tan   range is 60m dx 2 2 v x   2634 t cos 
98. 98. 1b) range c) velocity and direction of the ball after second 2  x 2 3x 1 when t  , x  10  x 2 3xy  y  80 4 2 80 4 x x  dy  x 3  3    4 20  dx 40 4 dy 1 1roots are 0 and 60 when x  10,  tan   range is 60m dx 2 2 v x   2634 t cos  5 1  2
99. 99. 1b) range c) velocity and direction of the ball after second 2  x 2 3x 1 when t  , x  10  x 2 3xy  y  80 4 2 80 4 x x  dy  x 3  3    4 20  dx 40 4 dy 1 1roots are 0 and 60 when x  10,  tan   range is 60m dx 2 2 v x   2634 t cos  5 1 10   2 1 2  2 5  10 5
100. 100. 1b) range c) velocity and direction of the ball after second 2  x 2 3x 1 when t  , x  10  x 2 3xy  y  80 4 2 80 4 x x  dy  x 3  3    4 20  dx 40 4 dy 1 1roots are 0 and 60 when x  10,  tan   range is 60m dx 2 2 v x   2634 t cos  5 1 10   2 1 2  2 5  10 5 1  after second, velocity  10 5m/s and it is 2 traveling at an angle of 26 34 to the horizontal
101. 101. 1b) range c) velocity and direction of the ball after second 2  x 3x 2 1 when t  , x  10  x 2 3x y  y  80 4 2 80 4 x x  dy  x 3  3    4 20  dx 40 4 dy 1 1roots are 0 and 60 when x  10,  tan    range is 60m dx 2 2 v x   2634 t cos  5 1 10 Exercise 3G; 1ac, 2ac,  2 1 24, 6, 8, 9, 11, 13, 16, 18  2 5 Exercise 3H; 2, 4, 6,  10 5 1 7, 10, 11  after second, velocity  10 5m/s and it is 2 traveling at an angle of 26 34 to the horizontal