1. Air Navigation Problems

2. Air Navigation Problems
There are three important “relative velocities” that pilots and air traffic controllers have to know. Do you know them?

3. Air Navigation Problems
Plane's velocity with respect to the ground or pvg
Plane's velocity with respect to the air or pva
The air molecule's velocity with respect to the ground or avg

4. Air Navigation Problems
Plane's velocity with respect to the ground or pvg
It is sometimes called “ground velocity”
|pvg| is sometimes called “ground speed” or pvg
It is used to calculate “flight time” using t=d/pvg
Plane's velocity with respect to the air or pva
The air molecule's velocity with respect to the ground or avg

5. Air Navigation Problems
Plane's velocity with respect to the ground or pvg
It is sometimes called “ground velocity”
|pvg| is sometimes called “ground speed” or pvg
It is used to calculate “flight time” using t=d/pvg
Plane's velocity with respect to the air or pva
The air molecule's velocity with respect to the ground or avg

6. Air Navigation Problems
Plane's velocity with respect to the air or pva

7. Air Navigation Problems
Plane's velocity with respect to the air or pva
It is really important. This velocity determines if the plane can “lift” off the ground or stay in the air. The |pva| is called “airspeed”. All planes have a |pva| or airspeed indicator in the cockpit.

8. Air Navigation Problems
Plane's velocity with respect to the ground or pvg
It is sometimes called “ground velocity”
|pvg| is sometimes called “ground speed” or pvg
It is used to calculate “flight time” using t=d/pvg
Plane's velocity with respect to the air or pva
It is really important. This velocity determines if the plane can “lift” off the ground or stay in the air. The |pva| is called “airspeed”. All planes have a |pva| or airspeed indicator in the cockpit.
The air molecule's velocity with respect to the ground or avg

9. Air Navigation Problems
Plane's velocity with respect to the ground or pvg
It is sometimes called “ground velocity”
|pvg| is sometimes called “ground speed” or pvg
It is used to calculate “flight time” using t=d/pvg
Plane's velocity with respect to the air or pva
It is really important. This velocity determines if the plane can “lift” off the ground or stay in the air. The |pva| is called “airspeed”. All planes have a |pva| or airspeed indicator in the cockpit.
The air molecule's velocity with respect to the ground or avg
This is just the speed and direction of the wind. “Wind” can blow a plane off course so we need to take it into account.

10. The Vector Chain Rule Equation for Air Navigation

11. The Vector Chain Rule Equation for Air Navigation
So far we have these relative velocities:
pvg and pva and avg
Can you use the chain rule to form a vector sum equation for pvg in terms of the other two relative velocities?
pvg = ??

12. The Vector Chain Rule Equation for Air Navigation
So far we have these relative velocities:
pvg and pva and avg
Can you use the chain rule to form a vector sum equation for pvg in terms of the other two relative velocities?
pvg = pva + avg

13. The Vector Chain Rule Equation for Air Navigation
So far we have these relative velocities:
pvg and pva and avg
Can you use the chain rule to form a vector sum equation for pvg in terms of the other two relative velocities?
pvg = pva + avg Memorize this air navigation equation!

14. Additional Coda for Air Navigation Velocities

15. Additional Coda for Air Navigation Velocities
The direction of pvg is called the “track” or “course”. This is the direction that the plane actually moves with respect to points on land.
It is not the direction that the pilot points the nose of the plane.

16. Additional Coda for Air Navigation Velocities
The direction of pvg is called the “track” or “course”. This is the direction that the plane actually moves with respect to points on land.
It is not the direction that the pilot points the nose of the plane.
The direction of pva is called the “heading” of the plane. It is the direction that the pilot points the nose of the plane.

17. Additional Coda for Air Navigation Velocities
The direction of pvg is called the “track” or “course”. This is the direction that the plane actually moves with respect to points on land.
It is not the direction that the pilot points the nose of the plane.
The direction of pva is called the “heading” of the plane. It is the direction that the pilot points the nose of the plane.
Why can't we assume pointing our plane with a heading of “north” will move us “north” w.r.t. the land?

18. Example #1: A pilot points her plane due south
Example #1: A pilot points her plane due south. The plane has an airspeed of km/h. A wind is blowing from the east at km/h. What is the plane's ground velocity? What is the plane's displacement relative to the ground after 8.00 h of travel?

19. Example #1: A pilot points her plane due south
Example #1: A pilot points her plane due south. The plane has an airspeed of km/h. A wind is blowing from the east at km/h. What is the plane's ground velocity? What is the plane's displacement relative to the ground after 8.00 h of travel?
Given:
pva = ?
avg = ?

20. Example #1: A pilot points her plane due south
Example #1: A pilot points her plane due south. The plane has an airspeed of km/h. A wind is blowing from the east at km/h. What is the plane's ground velocity? What is the plane's displacement relative to the ground after 8.00 h of travel?
Given:
pva = km/h [S]
avg = ?

21. Example #1: A pilot points her plane due south
Example #1: A pilot points her plane due south. The plane has an airspeed of km/h. A wind is blowing from the east at km/h. What is the plane's ground velocity? What is the plane's displacement relative to the ground after 8.00 h of travel?
Given:
pva = km/h [S]
avg = km/h [W]

22. Example #1: A pilot points her plane due south
Example #1: A pilot points her plane due south. The plane has an airspeed of km/h. A wind is blowing from the east at km/h. What is the plane's ground velocity? What is the plane's displacement relative to the ground after 8.00 h of travel?
Given:
pva = km/h [S]
avg = km/h [W]
Unknown:

23. Example #1: A pilot points her plane due south
Example #1: A pilot points her plane due south. The plane has an airspeed of km/h. A wind is blowing from the east at km/h. What is the plane's ground velocity? What is the plane's displacement relative to the ground after 8.00 h of travel?
Given:
pva = km/h [S]
avg = km/h [W]
Unknown:
pvg = ?

24. Example #1: A pilot points her plane due south
Example #1: A pilot points her plane due south. The plane has an airspeed of km/h. A wind is blowing from the east at km/h. What is the plane's ground velocity? What is the plane's displacement relative to the ground after 8.00 h of travel?
Given:
pva = km/h [S]
avg = km/h [W]
Unknown:
pvg = ?
Formula:

25. Example #1: A pilot points her plane due south
Example #1: A pilot points her plane due south. The plane has an airspeed of km/h. A wind is blowing from the east at km/h. What is the plane's ground velocity? What is the plane's displacement relative to the ground after 8.00 h of travel?
Given:
pva = km/h [S]
avg = km/h [W]
Unknown:
pvg = ?
Formula:
pvg = pva + avg

26. Example #1: A pilot points her plane due south
Example #1: A pilot points her plane due south. The plane has an airspeed of km/h. A wind is blowing from the east at km/h. What is the plane's ground velocity? What is the plane's displacement relative to the ground after 8.00 h of travel?
Given:
pva = km/h [S]
avg = km/h [W]
Unknown:
pvg = ?
Formula:
pvg = pva + avg
Substitiute:

27. Example #1: A pilot points her plane due south
Example #1: A pilot points her plane due south. The plane has an airspeed of km/h. A wind is blowing from the east at km/h. What is the plane's ground velocity? What is the plane's displacement relative to the ground after 8.00 h of travel?
Given:
pva = km/h [S]
avg = km/h [W]
Unknown:
pvg = ?
Formula:
pvg = pva + avg
Substitiute: pvg = ?

28. Example #1: A pilot points her plane due south
Example #1: A pilot points her plane due south. The plane has an airspeed of km/h. A wind is blowing from the east at km/h. What is the plane's ground velocity? What is the plane's displacement relative to the ground after 8.00 h of travel?
Given:
pva = km/h [S]
avg = km/h [W]
Unknown:
pvg = ?
Formula:
pvg = pva + avg
Substitiute: pvg = km/h [S] km/h [W]

29. Substitiute: pvg = 270.0 km/h [S] + 144.0 km/h [W] pvg
Example #1: A pilot points her plane due south. The plane has an airspeed of km/h. A wind is blowing from the east at km/h. What is the plane's ground velocity? What is the plane's displacement relative to the ground after 8.00 h of travel?
Given:
pva = km/h [S]
avg = km/h [W]
Unknown:
pvg = ?
Formula:
pvg = pva + avg
Substitiute: pvg = km/h [S] km/h [W]
pvg
270 km/h
144 km/h

30. Substitiute: pvg = 270.0 km/h [S] + 144.0 km/h [W] | pvg | = ? pvg
Example #1: A pilot points her plane due south. The plane has an airspeed of km/h. A wind is blowing from the east at km/h. What is the plane's ground velocity? What is the plane's displacement relative to the ground after 8.00 h of travel?
Given:
pva = km/h [S]
avg = km/h [W]
Unknown:
pvg = ?
Formula:
pvg = pva + avg
Substitiute: pvg = km/h [S] km/h [W]
| pvg | = ?
pvg
270 km/h
144 km/h

31. Substitiute: pvg = 270.0 km/h [S] + 144.0 km/h [W]
Example #1: A pilot points her plane due south. The plane has an airspeed of km/h. A wind is blowing from the east at km/h. What is the plane's ground velocity? What is the plane's displacement relative to the ground after 8.00 h of travel?
Given:
pva = km/h [S]
avg = km/h [W]
Unknown:
pvg = ?
Formula:
pvg = pva + avg
Substitiute: pvg = km/h [S] km/h [W]
| pvg | = ( )1/2 = ?
pvg
270 km/h
144 km/h

32. Substitiute: pvg = 270.0 km/h [S] + 144.0 km/h [W]
Example #1: A pilot points her plane due south. The plane has an airspeed of km/h. A wind is blowing from the east at km/h. What is the plane's ground velocity? What is the plane's displacement relative to the ground after 8.00 h of travel?
Given:
pva = km/h [S]
avg = km/h [W]
Unknown:
pvg = ?
Formula:
pvg = pva + avg
Substitiute: pvg = km/h [S] km/h [W]
| pvg | = ( )1/2 = 306 km/h
pvg
270 km/h
144 km/h

33. Substitiute: pvg = 270.0 km/h [S] + 144.0 km/h [W]
Example #1: A pilot points her plane due south. The plane has an airspeed of km/h. A wind is blowing from the east at km/h. What is the plane's ground velocity? What is the plane's displacement relative to the ground after 8.00 h of travel?
Given:
pva = km/h [S]
avg = km/h [W]
Unknown:
pvg = ?
Formula:
pvg = pva + avg
Substitiute: pvg = km/h [S] km/h [W]
| pvg | = ( )1/2 = 306 km/h
Where is the reference angle?
pvg
270 km/h
144 km/h

34. Example #1: A pilot points her plane due south
Example #1: A pilot points her plane due south. The plane has an airspeed of km/h. A wind is blowing from the east at km/h. What is the plane's ground velocity? What is the plane's displacement relative to the ground after 8.00 h of travel?
Given:
pva = km/h [S]
avg = km/h [W]
Unknown:
pvg = ?
Formula:
pvg = pva + avg
Substitiute: pvg = km/h [S] km/h [W]
| pvg | = ( )1/2 = 306 km/h
Θ = ?
pvg θ
270 km/h
144 km/h

35. Example #1: A pilot points her plane due south
Example #1: A pilot points her plane due south. The plane has an airspeed of km/h. A wind is blowing from the east at km/h. What is the plane's ground velocity? What is the plane's displacement relative to the ground after 8.00 h of travel?
Given:
pva = km/h [S]
avg = km/h [W]
Unknown:
pvg = ?
Formula:
pvg = pva + avg
Substitiute: pvg = km/h [S] km/h [W]
| pvg | = ( )1/2 = 306 km/h
Θ = tan-1(144/270) = ?
pvg θ
270 km/h
144 km/h

36. Example #1: A pilot points her plane due south
Example #1: A pilot points her plane due south. The plane has an airspeed of km/h. A wind is blowing from the east at km/h. What is the plane's ground velocity? What is the plane's displacement relative to the ground after 8.00 h of travel?
Given:
pva = km/h [S]
avg = km/h [W]
Unknown:
pvg = ?
Formula:
pvg = pva + avg
Substitiute: pvg = km/h [S] km/h [W]
| pvg | = ( )1/2 = 306 km/h
Θ = tan-1(144/270) = 28.1°
pvg θ
270 km/h
144 km/h

37. Example #1: A pilot points her plane due south
Example #1: A pilot points her plane due south. The plane has an airspeed of km/h. A wind is blowing from the east at km/h. What is the plane's ground velocity? What is the plane's displacement relative to the ground after 8.00 h of travel?
Given:
pva = km/h [S]
avg = km/h [W]
Unknown:
pvg = ?
Formula:
pvg = pva + avg
Substitiute: pvg = km/h [S] km/h [W]
| pvg | = ( )1/2 = 306 km/h
Θ = tan-1(144/270) = 28.1° so pvg = ?
pvg θ
270 km/h
144 km/h

38. Example #1: A pilot points her plane due south
Example #1: A pilot points her plane due south. The plane has an airspeed of km/h. A wind is blowing from the east at km/h. What is the plane's ground velocity? What is the plane's displacement relative to the ground after 8.00 h of travel?
Given:
pva = km/h [S]
avg = km/h [W]
Unknown:
pvg = ?
Formula:
pvg = pva + avg
Substitiute: pvg = km/h [S] km/h [W]
| pvg | = ( )1/2 = 306 km/h
Θ = tan-1(144/270) = 28.1° so pvg = 306 km/h [S28.1°W]
pvg θ
270 km/h
144 km/h

39. Example #1: A pilot points her plane due south
Example #1: A pilot points her plane due south. The plane has an airspeed of km/h. A wind is blowing from the east at km/h. What is the plane's ground velocity? What is the plane's displacement relative to the ground after 8.00 h of travel?

40. Example #1: A pilot points her plane due south
Example #1: A pilot points her plane due south. The plane has an airspeed of km/h. A wind is blowing from the east at km/h. What is the plane's ground velocity? What is the plane's displacement relative to the ground after 8.00 h of travel?
What equation gives us displacement in terms of constant velocity and time?

41. Example #1: A pilot points her plane due south
Example #1: A pilot points her plane due south. The plane has an airspeed of km/h. A wind is blowing from the east at km/h. What is the plane's ground velocity? What is the plane's displacement relative to the ground after 8.00 h of travel?
Δd = vΔt
= ?

42. Example #1: A pilot points her plane due south
Example #1: A pilot points her plane due south. The plane has an airspeed of km/h. A wind is blowing from the east at km/h. What is the plane's ground velocity? What is the plane's displacement relative to the ground after 8.00 h of travel?
Δd = vΔt
= 306 km/h [S28.1°W] X 8.00 h

43. Δd = vΔt = 306 km/h [S28.1°W] X 8.00 h = 2448 km [S28.1°W]
Example #1: A pilot points her plane due south. The plane has an airspeed of km/h. A wind is blowing from the east at km/h. What is the plane's ground velocity? What is the plane's displacement relative to the ground after 8.00 h of travel?
Δd = vΔt
= 306 km/h [S28.1°W] X 8.00 h
= km [S28.1°W]

44. Δd = vΔt = 306 km/h [S28.1°W] X 8.00 h = 2448 km [S28.1°W]
Example #1: A pilot points her plane due south. The plane has an airspeed of km/h. A wind is blowing from the east at km/h. What is the plane's ground velocity? What is the plane's displacement relative to the ground after 8.00 h of travel?
Δd = vΔt
= 306 km/h [S28.1°W] X 8.00 h
= km [S28.1°W]
Now try #3 on the air navigations problems handout for lesson #10 We will take it up in class.