Relative velocity question - plane and wind

[maskd]

Relative velocity question -- plane and wind

The distance between two cities A and B is l. An airplane travels back and forth between A and B, flying in a line, with velocity V relative to the air. (a) Find the total flight time, if the wind blows with velocity v, in a direction that forms an angle \Theta with the direction AB. Does this time depends on the direction where the wind blows? (b) Show that the whole travel is only possible if v < V, and find the relation between the flight time when the wind blows in the direction AB and the time when it blows in the perpendicular direction; (c) Show that, whatever the direction of the wind is, it always turns the travel duration longer.

Note: the original question is in another language -- I tried to make the translation as accurate as possible.

 

[member 553083]

a) The total flight time will depend on the direction of the wind. If the wind is blowing in the same direction as the airplane, the airplane will have an additional velocity of v relative to the ground, and the flight time will be decreased by a factor of V/(V+v). However, if the wind is blowing in the opposite direction, the airplane will have a velocity of V-v relative to the ground, and the flight time will be increased by a factor of (V+v)/V. b) The whole travel is only possible if v < V, since if v >= V the airplane will not be able to make any progress against the wind. The relation between the flight time when the wind blows in the direction AB and the time when it blows in the perpendicular direction is given by T_{AB}/T_{\perp} = (V+v)/(V-v), where T_{AB} is the flight time when the wind is blowing in the direction AB and T_{\perp} is the flight time when the wind is blowing in the perpendicular direction. c) Regardless of the direction of the wind, it always increases the travel duration. This is because the airplane will always have to fight the wind, and thus will take longer to reach its destination.