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.

 

[Nate810]

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.

 

[kyle8921]

(a) The total flight time can be calculated using the formula d=vt, where d is the distance between cities A and B, v is the velocity of the airplane relative to the air, and t is the total flight time. However, in this case, the wind is also a factor in the flight. The wind's velocity, v, and direction, \Theta, must be taken into account in order to accurately calculate the flight time. The formula for relative velocity, v_r = v_a - v_w, where v_r is the relative velocity, v_a is the velocity of the airplane, and v_w is the velocity of the wind, can be used to determine the total flight time. This time will indeed depend on the direction where the wind blows, as the wind's velocity and direction will affect the relative velocity of the airplane.

(b) In order for the airplane to successfully travel between cities A and B, the wind's velocity must be less than the airplane's velocity. This can be represented by the inequality v < V, where v is the wind's velocity and V is the airplane's velocity. If the wind blows in the direction AB, the flight time can be calculated using the formula t = d / (V - v). If the wind blows in the perpendicular direction, the flight time can be calculated using the formula t = d / (V^2 - v^2)^0.5. Therefore, the relation between the two flight times is t(perpendicular) = t(AB) / (V^2 - v^2)^0.5.

(c) Regardless of the direction of the wind, it will always add to the total flight time. This is because the wind's velocity and direction will affect the relative velocity of the airplane, resulting in a longer flight time. Therefore, no matter the direction of the wind, the travel duration will always be longer.