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

• lamp23
In summary, the conversation discusses the concept of adding vectors in plane-wind problems and the assumptions made about the wind and the objects moving in the medium. The example problem given involves a plane and a river, with the wind or current acting on them. The wind is assumed to have a constant velocity and direction, and the objects are affected by this force of the wind. The conversation also mentions the importance of considering the weight and mass of the objects when dealing with these types of problems. Overall, the conversation emphasizes the idea of the wind or medium as a force and its effect on the movement of objects.
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?

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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.

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.

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lamp23 said:
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.

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.

Don't underestimate wind:

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

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## 1. How do we define a vector in a Plane-Wind problem?

In a Plane-Wind problem, a vector represents the magnitude and direction of the wind's velocity. It is typically denoted by an arrow pointing in the direction of the wind's movement, with the length of the arrow representing the wind's speed.

## 2. Why is vector addition used in Plane-Wind problems?

Vector addition is used in Plane-Wind problems because it allows us to combine the wind's velocity with the velocity of the plane to determine the resulting speed and direction of the plane's movement. This is because vectors follow the principle of superposition, meaning that they can be added together to determine a net vector.

## 3. What is the process for adding vectors in Plane-Wind problems?

The process for adding vectors in Plane-Wind problems involves breaking down each vector into its x and y components, adding the corresponding components together, and then using the Pythagorean theorem and inverse tangent function to determine the magnitude and direction of the resulting vector.

## 4. Can we add more than two vectors in a Plane-Wind problem?

Yes, we can add more than two vectors in a Plane-Wind problem by repeating the process of adding the components and using the Pythagorean theorem and inverse tangent function for each additional vector. However, it is important to note that the order in which the vectors are added can affect the resulting vector.

## 5. What is the significance of the resulting vector in a Plane-Wind problem?

The resulting vector in a Plane-Wind problem represents the net velocity of the plane, taking into account both the plane's velocity and the wind's velocity. This can be used to determine the plane's speed and direction of movement, as well as to adjust the plane's course to compensate for the wind.

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