How Should Direction Be Adjusted in Relative Velocity Problems?

[Jacobpm64]

I have a couple of problems.. I'll show all my work...

A swimmer wants to swim due east in a stream that flows due south. In which direction should the swimmer swim?
The choices are east, north, northeast, south, southeast, and west... I chose Northeast.


Here's another question... use the bolded situation for the next 2 questions.

A plane flies west with a 100km/h velocity with respect to the air while the wind is blowing toward the north at 65km/h relative to the ground.

If the plane wants to fly directly west with respect to the ground, in what direction should he head his plane with respect to the air?
southwest? I'm not sure.. i think i need degrees for this too...

What if the wind is blowing directly south rather than north, where should the pilot head his plane if he wants to go directly west?
northwest? I'm not sure.. i think i need degrees for this too...

 

[Claude Bile]

Relative velocity problems are simple vector subtractions.

Diagramatically, you can represent the velocity of an object with a vector. The length of the vector represents the magnitude of the velocity, while the direction of the vector is simply the direction the object is travelling.

When adding vectors, simply place the vectors head to tail. The sum is the vector that joins the free 'tail' to the free 'head' (in that order, so the 'head' of the sum vector corresponds to the free head in the diagram). To subtract vectors, simply add a vector that is equal in magnitude but opposite in direction.

When you complete these subtractions, you get a triangle. From there it is a matter of using trigonometry to figure out magnitudes and directions.

Claude.

 

[quicknote]

As a scientist, it is important to understand the concept of relative velocity and how it applies to different situations. In the first scenario, the swimmer should actually swim southeast in order to reach their desired direction of due east. This is because the swimmer's velocity in the water will be added to the velocity of the water flow, resulting in a net velocity towards the southeast.

In the second scenario, the direction the plane should head in order to fly directly west with respect to the air will depend on the direction of the wind. If the wind is blowing towards the north, the pilot should head their plane in a southwest direction. This is because the wind's velocity will be subtracted from the plane's velocity, resulting in a net velocity towards the west and slightly south. If the wind is blowing towards the south, the pilot should head their plane in a northwest direction for the same reason.

It is not necessary to calculate the degrees in this situation, as the directions provided (southwest and northwest) are sufficient for the pilot to know where to head their plane. However, if the velocity of the wind and the plane were given, it would be possible to calculate the exact angle at which the plane should fly to achieve a net velocity of due west.