I was studying the chapter "circular motion" and there I studied the property-"the tangential component of acceleration of a particle undergoing a circular motion just happens to be equal to the instantaneous rate of change of speed of the particle" and it has been proved mathematically in the book. I was thinking about extension of this property to any planar trajectory of a particle.
1)If a particle moves in a plane in any unknown trajectory then at any instant the tangential component of the acceleration will be equal to the instantaneous rate of change of the speed of the particle. I want to know whether my claim is true or not. I have a reasoning in favour of my claim. Though it is not a mathematical one.
2)If we consider random trajectory of the particle,then at any point of the trajectory if we consider the osculating circle(the circle which best approximates the curve around that point) of the trajectory at that point then we can approximate the motion of the particle around that point as the motion of the particle along the osculating circle. So on the basis of this logic I am thinking that my claim is right.
I want to know is my claim right? If yes then is my reasoning a correct explanation for my claim? Can you please provide me a better explanation for my claim? A mathematical explanation is appreciated. Edit:- By tangential component of acceleration I want to mean the component of acceleration along the direction of velocity.