Trying to understand the logic behind adding vectors with an angle between them

[renobueno4153]

Homework Statement

Two aircrafts travel on the same plane with velocity v1 = 500 km/h and v2 = 800 km/h
respectively. The directions of motion generate an angle θ = 30◦ as the two aircrafts move
away from each other. Evaluate:

(a) The magnitude of the relative velocity of the second aircraft with respect to the first
one

Relevant Equations

law of cosine
vector addition

My initial calculation was to subtract V1 from V2 to show that from the perspective of the second aircraft the first one is -300km/h.

So i checked with ChatGPT and it said I cant just subtract them because I have an angle between them. So I dont understand the reasoning of it. Like why should a velocity be dependent on an angle? I was thinking about how it would look like if the planes where parallel to each other, and then how it look like if one is turning away and I dont see it. Since the planes just change direction but their speeds stay the same. With a lot of prompting with Chatgpt I got this sketch with the calculation out of it. And this just confuses me. Like why would I even want to know about the vector between them and then use the law of cosine. I mean I can see using the cosin in a sense of actually involving one of the given variables but intuitively I dont.

 

[PeroK]

Suppose you have a triangle with two sides that are 3 and 4 units respectively. Does the length of the third side depend on the angle between the first two sides?

 

[wrobel]

There is a regular theory which is good to know before solving such problems.
Banach Mech
You need formula (I) at page 57 of this book; and explanation of this formula above at page 56.

 

[kuruman]

Homework Statement: Two aircrafts travel on the same plane with velocity v1 = 500 km/h and v2 = 800 km/h
respectively.

Since the planes just change direction but their speeds stay the same.

Where does it say that the directions change?

 

[PeroK]

There is a regular theory which is good to know before solving such problems.
Banach Mech
You need formula (I) at page 57 of this book; and explanation of this formula above at page 56.

That source is perhaps a little advanced for the OP at this stage.

 

[kuruman]

Like why would I even want to know about the vector between them and then use the law of cosine.

Because, like, you are asked to find the relative velocity between the two planes. That is defined as the vector difference between the two velocities and is represented graphically as the arrow that you call $v_{21}$ in your drawing. Read about relative velocity here.

 

[gmax137]

Like why would I even want to know about the vector between them and then use the law of cosine.

@renobueno4153 I really hope you think about this until you understand it -- it holds the key idea in much of what follows in classical mechanics.

 

[Herman Trivilino]

My initial calculation was to subtract V1 from V2 to show that from the perspective of the second aircraft the first one is -300km/h.

So i checked with ChatGPT and it said I cant just subtract them because I have an angle between them.

In other words, you don't subtract the magnitudes of the vectors, you subtract the vectors.

This means you find the vector that you add to $\vec{v_1}$ to get $\vec{v_2}$.