Projectile Motion
Projectile Motion    y           x
Projectile Motion    y                    x
Projectile Motion    y                    x
Projectile Motion                     y       maximum range                                   45                       ...
Projectile Motion                     y            maximum range                                        45             ...
Projectile Motion                         y            maximum range                                            45     ...
Projectile Motion                         y                    maximum range                                              ...
Projectile Motion                         y                     maximum range                                             ...
Projectile Motion                         y                     maximum range                                             ...
Projectile Motion                         y                     maximum range                                             ...
  0x           g         y
  0x           g         yx  c1
  0x           g         yx  c1        y   gt  c2         
  0       x                    g                         y      x  c1                        y   gt  c2       ...
  0       x                    g                         y      x  c1                        y   gt  c2       ...
  0       x                      g                           y      x  c1                          y   gt  c2 ...
  0       x                        g                             y      x  c1                            y   gt...
  0       x                        g                             y      x  c1                            y   gt...
  0       x                        g                             y      x  c1                            y   gt...
  0       x                          g                               y      x  c1                             y ...
  0       x                                   g                                        y      x  c1              ...
  0            x                                      g                                                y           ...
  0          x                                     g                                             y         x  c1  ...
  0          x                                       g                                               y         x  ...
Common Questions
Common Questions(1) When does the particle hit the ground?
Common Questions(1) When does the particle hit the ground?    Particle hits the ground when y  0
Common Questions(1) When does the particle hit the ground?    Particle hits the ground when y  0(2) What is the range of ...
Common Questions(1) When does the particle hit the ground?    Particle hits the ground when y  0(2) What is the range of ...
Common Questions(1) When does the particle hit the ground?    Particle hits the ground when y  0(2) What is the range of ...
Common Questions(1) When does the particle hit the ground?    Particle hits the ground when y  0(2) What is the range of ...
Common Questions(1) When does the particle hit the ground?    Particle hits the ground when y  0(2) What is the range of ...
Common Questions(1) When does the particle hit the ground?    Particle hits the ground when y  0(2) What is the range of ...
Common Questions(1) When does the particle hit the ground?    Particle hits the ground when y  0(2) What is the range of ...
Common Questions(1) When does the particle hit the ground?    Particle hits the ground when y  0(2) What is the range of ...
Common Questions(1) When does the particle hit the ground?    Particle hits the ground when y  0(2) What is the range of ...
Common Questions(1) When does the particle hit the ground?    Particle hits the ground when y  0(2) What is the range of ...
Common Questions(1) When does the particle hit the ground?    Particle hits the ground when y  0(2) What is the range of ...
Common Questions(1) When does the particle hit the ground?    Particle hits the ground when y  0(2) What is the range of ...
Common Questions(1) When does the particle hit the ground?    Particle hits the ground when y  0(2) What is the range of ...
Summary
Summary A particle undergoing projectile motion obeys
Summary A particle undergoing projectile motion obeys              0            x           and            g       ...
Summary A particle undergoing projectile motion obeys               0             x             and         g      ...
Summary A particle undergoing projectile motion obeys               0             x             and          g     ...
Summary   A particle undergoing projectile motion obeys                  0                x            and           ...
Summary    A particle undergoing projectile motion obeys                  0                x             and          ...
Summary    A particle undergoing projectile motion obeys                   0                 x             and         ...
Summary    A particle undergoing projectile motion obeys                   0                 x             and         ...
Summary    A particle undergoing projectile motion obeys                   0                 x             and         ...
Summary    A particle undergoing projectile motion obeys                   0                 x             and         ...
Summary    A particle undergoing projectile motion obeys                   0                 x             and         ...
Summary    A particle undergoing projectile motion obeys                   0                 x             and         ...
  0x          10         y
  0x          10         yx  c1
  0x          10         yx  c1        y  10t  c2         
  0    x                  10                     y   x  c1                    y  10t  c2                     ...
  0    x                  10                     y   x  c1                    y  10t  c2                     ...
  0    x                  10                     y   x  c1                    y  10t  c2                     ...
  0    x                  10                     y   x  c1                    y  10t  c2                     ...
  0    x                   10                      y   x  c1                    y  10t  c2                    ...
  0    x                   10                      y   x  c1                    y  10t  c2                    ...
  0    x                   10                      y   x  c1                    y  10t  c2                    ...
  0    x                    10                       y   x  c1                      y  10t  c2                ...
  0          x                             10                                      y         x  c1               ...
  0          x                             10                                      y         x  c1               ...
  0          x                                  10                                           y         x  c1      ...
  0          x                                   10                                            y         x  c1    ...
b) range
b) range  time of flight is 3 seconds
b) range  time of flight is 3 seconds  when t  3, x  203                 60
b) range  time of flight is 3 seconds  when t  3, x  203                 60   range is 60m
b) range   time of flight is 3 seconds   when t  3, x  203                  60   range is 60m                       ...
b) range   time of flight is 3 seconds   when t  3, x  203                  60   range is 60m                       ...
b) range   time of flight is 3 seconds   when t  3, x  203                  60   range is 60m                       ...
b) range   time of flight is 3 seconds   when t  3, x  203                  60   range is 60m                       ...
b) range   time of flight is 3 seconds   when t  3, x  203                  60   range is 60m                       ...
d) cartesian equation of the path
d) cartesian equation of the path        x  20t             x        t            20
d) cartesian equation of the path        x  20t              y  5t 2  15t                                         2   ...
d) cartesian equation of the path        x  20t              y  5t 2  15t                                         2   ...
d) cartesian equation of the path        x  20t              y  5t 2  15t                                         2   ...
d) cartesian equation of the path         x  20t              y  5t 2  15t                                          2 ...
d) cartesian equation of the path         x  20t               y  5t 2  15t                                           ...
d) cartesian equation of the path         x  20t               y  5t 2  15t                                           ...
d) cartesian equation of the path         x  20t               y  5t 2  15t                                           ...
d) cartesian equation of the path         x  20t               y  5t 2  15t                                           ...
b) range
b) range    x 2 3xy         80     4   x      x   3    4     20 
b) range    x 2 3xy         80     4   x      x   3    4     20 roots are 0 and 60 range is 60m
1b) range             c) velocity and direction of the ball after second                                                  ...
1b) range             c) velocity and direction of the ball after second                                                  ...
1b) range             c) velocity and direction of the ball after second                                                  ...
1b) range             c) velocity and direction of the ball after second                                                  ...
1b) range             c) velocity and direction of the ball after second                                                  ...
1b) range             c) velocity and direction of the ball after second                                                  ...
1b) range             c) velocity and direction of the ball after second                                                  ...
1b) range             c) velocity and direction of the ball after second                                                  ...
1b) range             c) velocity and direction of the ball after second                                                  ...
1b) range                  c) velocity and direction of the ball after second                                             ...
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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

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