# What is headwind and tailwind

## Superimposed speeds

 taskA plane flies 100 km against the wind in a constant wind and 100 km back with a tailwind! Can the aircraft compensate for the time lost due to the headwind on the return flight with the tailwind?

### Explanations

• If the aircraft flies at wind speed (over the ground) in the limit, it takes an infinitely long time until it arrives against the wind. On the return flight, the time is only halved by doubling the speed. [1]
• Perhaps one can argue like this: With a tailwind you gain distance, with a headwind you lose distance, and the same amount per hour. The path gained is spared, the lost path is reworked. Since you are on the road longer in a headwind, you lose more distance than you gain in a tailwind.
v: normal aircraft speed v-w: with headwind v + w: with (same) tailwind t1: time with headwind t2: time with tailwind

Because the movement with a headwind takes longer, the "v-w" and the "v + w" components must be weighted with their times when determining the average speed, so the average speed vm is then below v and the flight time is always greater.

### Invoice:

vm = total distance / total time vm = (t1 * (vw) + t2 * (v + w)) / (t1 + t2)) = ... = v - w * (t1 - t2) / (t1 + t2) < v, since t1> t2

### Simple example

Distance 1000 km; Airplane 500 km / h;
without wind: 500 km / h against the ground - 2 hours
Wind 100 km / h
with tail wind: 600 km / h against the ground - 5/3 h - 1/3 h = 20 min gain
with headwind: 400 km / h against the ground - 5/2 h - 1/2 h = 30 min loss
Wind 250 km / h
with tail wind: 750 km / h against the ground - 4/3 h - 2/3 h = 40 min gain
with headwind: 250 km / h against the ground - 4 h - 2 h = 120 min loss
Wind 400 km / h
with tail wind: 900 km / h against the ground - 10/9 h - 8/9 h = 53 1/3 min
Profit
with headwind: 100 km / h against the ground - 10 h - 8 h = 480 min loss
Wind 500 km / h
with tail wind: 1000 km / h against the ground - 1 h - = 60 min profit
with headwind: 0 km / h against the ground does not arrive

(It is also "nice" that you can do everything in your head with fractions, but not with decimal numbers.)

### swell

1. ↑ The example probably comes from Gerthsen:
Gerthsen Physik, series: Springer-Lehrbuch, Meschede, Dieter (Ed.), 23., revised. Ed., 2006, XX, 1162 S., 1347 illus., Geb., ISBN 978-3-540-25421-8