I am a maths undergraduate. I decided to start studying some basic physics trying to be rigorous with the math and uderstand the interpretations of some of the mathematical constructs and see how they are modeled to explain physics. Vectors and vector spaces are all well defined in math.
So the first interpretation that I stumble upon is seeing vectors as "arrows" with length and direction and that these arrows follow the rules of vectors (that was done also in math calculus textbooks which I think it might be confusing if you are trying to be rigorous).
At first it was ok. But now on chapter 4 (Halliday Resnick physics) on circular motion, I found that it is an accelerating motion with constant velocity which i find it very counter intuitive since if you are accelerating means you have to be speeding up, now this is a result of the the vector interpretation giving vectors directions and the direction of the velocity is changing so i have acceleration. Still the acceleration is the rate of change of the velocity with time. And if you told me acceleration is constant I would integrate to find the velocity but that doesnt seem to work on circular motion. Why is that? Trying to understand it so that it doesnt seem counter intuitive to me. It is like if you are turning something makes you loose speed so in order to keep it constant you have to accelerate every time.Is that the case? but if it is, I want a more mathematical way to see it. Sometimes it feels like these "arrows" are not so good to explain things. Sorry if the question is not appropriate for the site.
(Halliday has not mentioned any forces so far and has explained things only using basic definitions as rate of change for velocity and acceleration and has done a mathematical trick to prove the acceleration of circular motion with similar triangles and some limits.)