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

x
Initial conditions

6. Projectile Motion
y maximum range
  45

x
Initial conditions when t = 0
v


7. Projectile Motion
y maximum range
  45

x
Initial conditions when t = 0
v 
y


x

8. Projectile Motion
y maximum range
  45

x
Initial conditions when t = 0

x
v  cos

y v
 x  v cos


x

9. Projectile Motion
y maximum range
  45

x
Initial 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

x
Initial 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

x
Initial conditions when t = 0

x 
y
v  cos  sin 

y v v
 x  v cos
 y  v sin 


x x0 y0

12.   0
x    g
y

13.   0
x    g
y
x  c1


14.   0
x    g
y
x  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
x
t
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;.
4
a) 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 obtained
Initial 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 obtained
Initial 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 obtained
Initial 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 obtained
Initial 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 obtained
Initial 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 obtained
Initial 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.   0
x   10
y

56.   0
x   10
y
x  c1


57.   0
x   10
y
x  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  15t
greatest 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  15t
greatest 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  15t
greatest 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  15t
greatest 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
1
c) 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
1
c) 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
1
c) 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
1
c) 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
1
c) 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 4
Using 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 4
Using the cartesian equation to solve the problem
a) 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 4
Using the cartesian equation to solve the problem
a) 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 4
Using the cartesian equation to solve the problem
a) 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 4
Using the cartesian equation to solve the problem
a) 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 3x
y 
80 4
x x 
 3 
4 20 

91. b) range
 x 2 3x
y 
80 4
x x 
 3 
4 20 
roots are 0 and 60
 range is 60m

92. 1
b) range c) velocity and direction of the ball after second
2
 x 2 3x
y 
80 4
x x 
 3 
4 20 
roots are 0 and 60
 range is 60m

93. 1
b) range c) velocity and direction of the ball after second
2
 x 2 3x 1
when t  , x  10
y 
80 4 2
x x 
 3 
4 20 
roots are 0 and 60
 range is 60m

94. 1
b) range c) velocity and direction of the ball after second
2
 x 2 3x 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
roots are 0 and 60
 range is 60m

95. 1
b) range c) velocity and direction of the ball after second
2
 x 2 3x 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
roots are 0 and 60 dy 1
when x  10, 
 range is 60m dx 2

96. 1
b) range c) velocity and direction of the ball after second
2
 x 2 3x 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 1
roots are 0 and 60 when x  10,  tan  
 range is 60m dx 2 2
  2634

97. 1
b) range c) velocity and direction of the ball after second
2
 x 2 3x 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 1
roots are 0 and 60 when x  10,  tan  
 range is 60m dx 2 2
v
x   2634
t cos 

98. 1
b) range c) velocity and direction of the ball after second
2
 x 2 3x 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 1
roots are 0 and 60 when x  10,  tan  
 range is 60m dx 2 2
v
x   2634
t cos  5
1

2

99. 1
b) range c) velocity and direction of the ball after second
2
 x 2 3x 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 1
roots 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. 1
b) range c) velocity and direction of the ball after second
2
 x 2 3x 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 1
roots 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. 1
b) 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 1
roots 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 2
4, 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