This very simple question was posed by a high-school student in the class.

Consider a particle going in a uniform circular motion (**uniform** implies that the speed is constant). We know that there is a centripetal acceleration at every point in the motion, which changes the direction of the velocity. 

Now, consider an infinitesimal period of time *dt* for this motion at some given instant. Let's say that the initial velocity is $v\_1\vec{i}$ in X-direction. Then, the acceleration will add up a component $a.dt\vec{j}$ to this velocity. Adding these 2 quantities should give us our final velocity $\vec{v\_2}$. See this diagram:

[![enter image description here](https://i.stack.imgur.com/VtZtO.png)](https://i.stack.imgur.com/VtZtO.png)

Now, it is clear from this reasoning that $|\vec{v\_2}| > v\_1$. So, the magnitude of the velocity should increase!! 

Now, I know that this is WRONG. I just don't know where the fault lies. Clearly, there is some fault in the mathematical modeling done here. 

I am thinking that there must be some fault in the causation modeling. Acceleration is not CAUSING the change in velocity. It is an EFFECT of the change in velocity. That is one probably answer. But I'd like you to shed more light on this conundrum. How should one go about explaining this to a high-school kid?