Airline pilots who fly round trips know that their round-trip travel time increases if there is any wind. To see this, suppose that an airliner cruises at speed \(v\) relative to the air.

(a) For a flight whose one-way distance is \(d\), write an expression for the interval \(\Delta t_{\mathrm{calm}}\) needed for a round trip on a windless day. Ignore any time spent on the ground, and assume that the airliner flies at cruising speed for essentially the whole trip.

(b) Now assume there is a wind of speed \(w\). It doesn't matter which way the wind is blowing; all that matters is that it is a head wind in one direction and a tail wind in the opposite direction. Write an expression for the time interval \(\Delta t_{\text {wind }}\) needed for a round trip on the day this wind is blowing. Then show that your expression reduces to 

provided that \(w \ll v\). (For the last step, you may need to know that \((1 \pm z)^{b} \approx 1 \pm b z\) when \(z \ll 1\).)

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Alwindfal