A particle undergoing projectile motion obeys two equations of motion: its horizontal acceleration is 0 and its vertical acceleration is -g. The particle has initial conditions when t=0 of an initial horizontal velocity of vcosθ and initial vertical velocity of vsinθ, where v is the initial speed and θ is the launch angle. For example, this can be used to model a ball thrown with an initial speed of 25 m/s at an angle of tan-1(3/4) to the ground to determine properties of its motion.

1. Projectile Motion

2. Projectile Motion
y
x

3. Projectile Motion
y
/s
vm

x

4. Projectile Motion
y
/s
vm

x

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 x0

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 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. Common Questions

23. Common Questions
(1) When does the particle hit the ground?

24. Common Questions
(1) When does the particle hit the ground?
Particle hits the ground when y  0

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

35. Summary

36. Summary
A particle undergoing projectile motion obeys

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

48.   0
x   10
y

49.   0
x   10
y
x  c1


50.   0
x   10
y
x  c1
 y  10t  c2


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 y0

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 y0
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 y0
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 y0
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 y0
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 y0
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 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

63. b) range

64. b) range
time of flight is 3 seconds

65. b) range
time of flight is 3 seconds
when t  3, x  203
 60

66. b) range
time of flight is 3 seconds
when t  3, x  203
 60
 range is 60m

67. 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

68. 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
when t  , x  20  1   15
 y  10 

2 2
 10

69. 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
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  203
 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
  2634

71. 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
when t  , x  20  1   15
 y  10 
 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

72. d) cartesian equation of the path

73. d) cartesian equation of the path
x  20t
x
t
20

74. d) cartesian equation of the path
x  20t y  5t 2  15t
2
x
y  5   15 
t x x
   
20  20   20 
 x 2 3x
y 
80 4

75. d) cartesian equation of the path
x  20t y  5t 2  15t
2
x
y  5   15 
t x x
   
20  20   20 
 x 2 3x
y 
80 4
Exercise 3G; 1ac, 2ac, 4, 6, 8, 9, 11, 13, 16, 18
Exercise 3H; 2, 4, 6, 7, 10, 11