Here’s a fun physics question for you. It goes like this:

Suppose a plane can fly with a constant air speed from point A to B and back without any wind. If there is a constant wind, what happens to the round trip time? Does it stay the same, increase, or decrease? Another option: it depends on the wind speed.

Here’s a picture to go along with this.

It’s fun because everyone probably has an idea for an answer and you don’t really need to get out a calculator to justify your choice (but you are welcome to crunch some numbers also).

Are you ready for the answer? If so, I’m going to give you three versions: a conceptual answer, an algebraic answer, and a python-based answer. Let’s go.

# Relative Velocity

We need to address one important point before getting to a solution. There’s a difference between a plane’s air speed and the ground speed. The air speed is the velocity of the plane with respect to the air. However, if the air is also moving (a thing we like to call “wind”) then the plane can have a different ground speed.

If the velocity of the plane with respect to the air is v_p-a and the velocity of the air with respect to the ground is v_a-g, then the ground speed (v_p-g) will be:

Yes, these are vectors — direction matters. Here’s a quick example. Suppose a plane has an airspeed of 100 km/hr and is flying in the +x direction with a 10 km/hr wind also in the +x direction. This plane would have a ground speed of 100+10 = 110 km/hr in the x direction.

# Conceptual Answer

One way to find an answer to this question is to consider the extreme cases. Suppose that the plane can fly with an airspeed of 100 km/hr and the distance between point A and B is 100 kilometers. If there is no wind, then the ground speed is also 100 km/hr. This means that going from point A to B will take 1 hour and from B to A will also take an hour for a round trip time of 2 hours. Simple. Right?