5. Projectile Motion
y maximum range
45
/s
vm
x
Initial conditions
6. Projectile Motion
y maximum range
45
/s
vm
x
Initial conditions when t = 0
v
7. Projectile Motion
y maximum range
45
/s
vm
x
Initial conditions when t = 0
v
y
x
8. Projectile Motion
y maximum range
45
/s
vm
x
Initial conditions when t = 0
x
v cos
y v
x v cos
x
9. Projectile Motion
y maximum range
45
/s
vm
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
/s
vm
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
/s
vm
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
25. 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?
26. 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
27. 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
28. 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?
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?
i find when y 0
ii substitute into x
(3) What is the greatest height of the particle?
i find when y 0
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
ii substitute into x
(3) What is the greatest height of the particle?
i find when y 0
ii substitute into y
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
(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?
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
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
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?
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
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
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
37. Summary
A particle undergoing projectile motion obeys
0
x and g
y
38. Summary
A particle undergoing projectile motion obeys
0
x and g
y
with initial conditions
39. Summary
A particle undergoing projectile motion obeys
0
x and g
y
with initial conditions
x v cos
and y v sin
40. 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
41. 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
42. 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
43. 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
44. 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
45. 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
4
x 25
tan 1
3
5 4
20m/s 4
46. 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
4
x 25
tan 1
3
5 4
20m/s 4
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
a) greatest height obtained
Initial conditions x v cos
y v sin
5
3
4 3
x 25
y 25
tan 1
3
5 5 4
20m/s 15m/s 4
51. 0
x 10
y
x c1
y 10t c2
when t 0, x 20
y 15
52. 0
x 10
y
x c1
y 10t c2
when t 0, x 20
y 15
c1 20
x 20
53. 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
54. 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
55. 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
56. 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
57. 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
58. 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
59. 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
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
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
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 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
2
3 15 3
10t 15 0 when t , y 5
2 2 2
3 45
t
2 4
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
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
65. b) range
time of flight is 3 seconds
when t 3, x 203
60
66. b) range
time of flight is 3 seconds
when t 3, x 203
60
range is 60m
67. 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
68. 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
when t , x 20 1 15
y 10
2 2
10
69. 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
when t , x 20 1 15
y 10
10 5
2 2 10
10
20
70. 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
when t , x 20 1 15
y 10
10 5
2 2 10
10
1
tan 20
2
2634
71. 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
when t , x 20 1 15
y 10
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