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 x0
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 x0 y0
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 y0
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 y0
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 y0
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 y0
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 y0
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 y0
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 y0
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
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
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
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 y0
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 y0
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 y0
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 y0
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 y0
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 y0
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 y0
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
72. b) range
time of flight is 3 seconds
when t 3, x 203
60
73. b) range
time of flight is 3 seconds
when t 3, x 203
60
range is 60m
74. b) range
time of flight is 3 seconds
when t 3, x 203
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 203
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 203
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 203
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
2634
78. b) range
time of flight is 3 seconds
when t 3, x 203
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
2634
1
after second, velocity 10 5m/s and it is traveling at
2
an angle of 2634 to the horizontal
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
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
2634
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 2634
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 2634
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 2634
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 2634
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 2634
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