## general puzzles

easy | general |

An airplane flies in a straight line from airport $A$ to airport $B$, and then back in a straight line from $B$ to $A$. It travels with a constant engine speed and there is no wind. Will the total travel time for the same round trip be greater, less or the same if, throughout both the flights a constant wind blows from $A$ to $B$?

Wind will always slow down the journey!

The more the wind blows, the total time will be higher even though one of the journeys is faster.

### Intuition

The time during which the plane's speed is boosted is shorter than the time during which it is slowed down, so the over all effect is of slow-down.

### Maths

Suppose the wind blows at a speed of $w$ from $A$ to $B$, and the plane's engine is capable of a steady speed of $s$ (without any wind). Ignore acceleration and de-acceleration.

Suppose the distance is $d$, the total time will be $2d/s$ without any wind.

Speed from $A$ to $B$ is $(s + w)$ and $B$ to $A$ is $(s - w)$

$t = \dfrac{d}{s+w} + \dfrac{d}{s-w}$

$= d ( \dfrac{2 s}{s^2 - w^2})$

$= \dfrac{2d}{s - (w^2/s) } > \dfrac{2d}{s}$

For $0 < w < s$, the denominator in the first term is less than $s$, and therefore the total time is greater than the time without the wind.

### Footnotes

- Consider the edge case, where the plane's speed is the same as the wind. In one direction, the plane's speed is doubled, while in the other, it becomes zero! (i.e. it takes an infinite amount of time to finish the journey!)