When a car is driving along a level turn, the only thing keeping it in its lane is friction. This must be large enough to provide the centripetal force—the turn is only safe if `vec F_"f" >= vec F_"c"`, which limits speed by `v <= sqrt(rmug)`.

Instead of relying on friction, we can incline the road towards the centre of the curve; this is called a banked turn. A banked turn with friction is pretty complicated, so we will consider *frictionless* banked turns. In this case, the normal force has to provide the centripetal force by itself.

The normal force can be split into vertical and horizontal components:

`F_"N,v" = |vec F_"N"|cos theta qquad and qquad F_"N,h" = |vec F_"N"|sin theta`.

The horizontal component must supply the centripetal force, so we can set them equal:

`|vec F_"N"|sin theta = (mv^2)/r`.

Since the vehicle doesn’t move vertically, the vertical component has to balance gravity with `|vec F_"N"|cos theta = mg`. We can solve this for the normal force and substitute it into our first equation to get

`(mg)/(cos theta)sin theta = (mv^2)/r`,

and solving for speed gives us the maximum safe speed on a frictionless banked curve,

`v = sqrt(rg tan theta)`.

We can also find the minimum safe radius at a given speed:

`r = v^2/(g tan theta)`.