# Why does the vector in the banked turn problem not point to the center?

I am currently working on the banked turn problem. I know that once you set up all the equations properly you end up with $$v^2=gR\tan{\theta}$$

I drew a little diagramm of the forces:

I know that the right way to do it is to separate the Normal force $$N$$ into the two lighblue components $$\color{blue}{N\sin{\theta}=\frac{mv^2}{R}}$$ $$\color{blue}{N\cos{\theta}=mg}$$ and after dividing the equations I arrive at the relation $$v^2=gR \tan{\theta}$$

My question is this: Why do we separate into the blue components and not into something containing the red component. In other words, isn't the red component the one pointing in the direction of the centripetal force $$\frac{mv^2}{R}$$?

• Thank you for your answer. If I understand what you said correctly then if I were to separate N into components containing the red vector then I couldn't have said $\color{red}{\text{Force component in }x \space \text{direction}}=\frac{mv^2}{r}$ because the red vector actually is not entirely in the direction of the centripetal force right? Nov 15, 2018 at 14:13