12 x1 t07 01 projectile motion (2012)

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  • 1. Projectile Motion
  • 2. Projectile Motion y x
  • 3. Projectile Motion y  x
  • 4. Projectile Motion y  x
  • 5. Projectile Motion y maximum range   45  xInitial conditions
  • 6. Projectile Motion y maximum range   45  xInitial conditions when t = 0 v 
  • 7. Projectile Motion y maximum range   45  xInitial conditions when t = 0 v  y   x
  • 8. Projectile Motion y maximum range   45  xInitial conditions when t = 0  x v  cos  y v  x  v cos   x
  • 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. 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. 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.   0x    g y
  • 13.   0x    g yx  c1
  • 14.   0x    g yx  c1 y   gt  c2 
  • 15.   0 x    g y x  c1  y   gt  c2 when t  0, x  v cos  y  v sin  
  • 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.   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.   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.   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.   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.   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.   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.   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.   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.   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. Common Questions
  • 27. Common Questions(1) When does the particle hit the ground?
  • 28. Common Questions(1) When does the particle hit the ground? Particle hits the ground when y  0
  • 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Summary
  • 43. Summary A particle undergoing projectile motion obeys
  • 44. Summary A particle undergoing projectile motion obeys   0 x and    g y
  • 45. Summary A particle undergoing projectile motion obeys   0 x and    g y with initial conditions
  • 46. Summary A particle undergoing projectile motion obeys   0 x and    g y with initial conditions x  v cos  and y  v sin  
  • 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. 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. 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. 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. 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. 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. 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. 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.   0x   10 y
  • 56.   0x   10 yx  c1
  • 57.   0x   10 yx  c1 y  10t  c2 
  • 58.   0 x   10 y x  c1  y  10t  c2 when t  0, x  20  y  15 
  • 59.   0 x   10 y x  c1  y  10t  c2 when t  0, x  20  y  15  c1  20 x  20 
  • 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.   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.   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.   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.   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.   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.   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.   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.   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.   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. b) range
  • 71. b) range time of flight is 3 seconds
  • 72. b) range time of flight is 3 seconds when t  3, x  203  60
  • 73. b) range time of flight is 3 seconds when t  3, x  203  60  range is 60m
  • 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. 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. 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. 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. 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. d) cartesian equation of the path
  • 80. d) cartesian equation of the path x  20t x t 20
  • 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. 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. 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. 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. 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. 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. 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. 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. b) range
  • 90. b) range  x 2 3xy  80 4 x x   3  4 20 
  • 91. b) range  x 2 3xy  80 4 x x   3  4 20 roots are 0 and 60 range is 60m
  • 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. 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. 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. 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. 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. 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. 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. 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. 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. 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