The following three questions all pertain to flights between Paris and Copenhage

Question

The following three questions all pertain to flights between Paris and Copenhagen, see figure below.

Copenhagen is located 780 km North and 810 km East of Paris. A flight from Paris to Copenhagen takes two hours. Typically the wind blows from East to West over Europe, but the speed varies. For one particular flight, the wind speed was 210 km/h.

What is the magnitude of v pa, the plane's velocity with respect to the air?

What is 1, the direction of v pa with respect to East?

Another day, the speed of the wind has changed to 231 km/h, still the Paris to Copenhagen flight time is two hours. How does the 2, the direction of v pa,2 on this day, compare to 1?

2 < 1

2 = 1

2 > 1

Explanation / Answer

Speed of wind wrt Ground Vwg = - 210 km/hr i
Poistion of Copenhagen from paris = 810 i + 780 j Km
Distance of Copenhagen from paris = sqrt(810^2 +780^2) = 1125 Km
Time Taken by flight to journey = 2 hr
Velocity of Flight wrt ground Vpg = 405 i + 390 j
Velocity of Flight wrt air = Vpa

Vpg = Vpa + Vwg
405 i + 390 j = Vpa - 210 i
Vpa = 615 i + 390 j
Magnitude of Velocity of Flight wrt ground |Vpa|= sqrt(615^2 + 390^2)
Magnitude of Velocity of Flight wrt ground |Vpa| = 728.25 km/hr

1 = tan^-1(390/615)
1 = 32.38o

Now, Speed of wind changes to 231 km/hr -
Vpg = Vpa + Vwg
405 i + 390 j = Vpa - 231 i
Vpa = 636 i + 390 j

2 = tan^-1(390/636)
2 = 31.52o
Thus we can see 2 < 1

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