How do we know we can just add vectors in Plane-Wind problems?

[lamp23]

In plane-wind problems where you are given the velocity of the boat and the velocity of the wind, my textbook gave the solution as just adding the vectors. How do we know we can just add the vectors? Even though the wind might be blowing faster than my boat, won't my boat still make progress even if it's heading straight into the wind? What is being assumed in these problems about the masses or the forces that I'm missing?

Here is an example problem:
The problem: A plane leaves the airport on a bearing of 45 degrees traveling at 400mph. The Wind is blowing at a bearing of 135 degrees at a speed of 40mph. What is the actual velocity of the plane?

 

[Pengwuino]

The basic idea is that you're traveling in the medium. Imagine the airplane doesn't have it's engines running. Of course, you'll need to pretend it's not going to just plummet to the ground. Assuming it doesn't fall, the airplane, in a medium (air) of 40mph at 135 degrees will eventually be sped up by the wind to 40mph at that bearing (and actually I always assumed wind "bearing at 135 degrees" meant the wind was blowing in the SE direction but whatever). So when you turn on the engines and point your airplane at 45 degrees and pick up your speed to 400mph, you must add the velocity contribution from the airplane itself plus the fact that you're in a medium which is moving the airplane.

The assumptions to the problem are that the airplane is immediately and constantly being given a 40mph/135 degree velocity due to the wind. Now, in reality, even if you assume the wind is constant everywhere, the airplane would take some time to reach that 40mph/135 degree velocity simply from it's own inertia fighting the wind trying to push it. However, this time is very short and can be neglected.

If the airplane is a little big unrealistic, think of a boat in a river. Put the boat at rest and allow the river to act on it. The boat will reach the river's speed VERY quickly. Same idea with an airplane, it just takes a bit longer.

 

[lamp23]

I wish they would state what they meant by the wind having such a speed. Otherwise I just think of a boat that is only traveling a few mph against a wind with a greater speed yet the boat still making progress, because I think of the wind not being nearly as massive as the boat and not creating a large enough force to move it.

 

[Pengwuino]

I wish they would state what they meant by the wind having such a speed. Otherwise I just think of a boat that is only traveling a few mph against a wind with a greater speed yet the boat still making progress.

They do say that it has a 40mph velocity and give it's direction.

 

[lamp23]

Hmm, I guess my main problem is I'm thinking of wind as a very lightweight object and comparing it to the plane or the boat as a much heavier object, but I guess I'm not supposed to be thinking of the wind as an object since there is a steady stream of it.
It's easier for me to think about the river obeying these laws because I'm not confusing it with something I could draw a force diagram for.
Thanks a lot for your help though, I definitely started thinking about it differently once you said to think of what would happen if the plane could just float in the air.

 

[Pengwuino]

Don't underestimate wind:



Notice the landing, he has to point into the wind substantially to continue going straight.