Q- The distance from A to B is l. A plane flies at constant speed v relative to air along straight line from A to B & then from B to A. Calculate total time taken for his round trip, if a wind is flowing in three directions of wind velocity (u)-

I)along the line ATO B
II)perpendicular to line AB
III) At an angle Ѳ to AB

To calculate the total time taken for the round trip, we need to consider the effects of the wind in three different scenarios: along the line AB, perpendicular to the line AB, and at an angle θ to AB.

I) When the wind is blowing along the line AB:
In this case, the plane's speed relative to the ground is the sum of its speed relative to the air (v) and the wind's speed along the line AB (u). The effective speed of the plane is v + u.
The time taken to travel from A to B is l divided by the effective speed: t1 = l / (v + u).
Similarly, the time taken to travel from B to A is also l divided by the effective speed: t2 = l / (v + u).
Therefore, the total time taken for the round trip is the sum of t1 and t2: t_total = t1 + t2 = l / (v + u) + l / (v + u).

II) When the wind is blowing perpendicular to the line AB:
In this case, the wind's velocity does not affect the plane's speed along the line AB. So, the time taken to travel from A to B and from B to A remains the same as when there is no wind: t1 = l / v and t2 = l / v.
Therefore, the total time taken for the round trip in this case is: t_total = t1 + t2 = l / v + l / v = 2l / v.

III) When the wind is blowing at an angle θ to the line AB:
In this case, the wind's velocity can be resolved into two components: one along the line AB and one perpendicular to the line AB. Let's call these components u1 and u2, respectively.
The effective speed of the plane while flying along the line AB is v + u1, as explained in scenario I.
The effective speed of the plane while flying perpendicular to the line AB is u2, as explained in scenario II.
Therefore, the total time taken for the round trip when the wind is blowing at an angle θ is: t_total = (l / (v + u1)) + (l / u2).

Now you can use these formulas to calculate the total time taken for the round trip in each scenario by plugging in the values of l, v, u, θ, u1, and u2.