Phase 1: Identification of Forces. The forces acting on a vehicle traversing a banked curve include:
- Gravitational force (\( mg \)).
- Normal reaction force (\( N \)).
- Frictional force (\( f \)).
The resultant force dictates the centripetal force required for circular motion.
Phase 2: Force Decomposition. The centripetal force is supplied by the component of the normal reaction force directed towards the center of the curve:
\[
N \sin \theta = \frac{m v^2}{r},
\]
wherein \( v \) represents velocity, \( r \) denotes the curve radius, and \( \theta \) signifies the banking angle.
Phase 3: Idealized Scenario (Maximum Safe Velocity). In a frictionless scenario, the component of the normal force satisfies the centripetal force requirement:
\[
v^2 = r g \tan \theta.
\]
The expression for \( v \) is:
\[
v = \sqrt{r g \tan \theta}.
\]
Phase 4: Conclusion. The maximum safe velocity is:
\[
\boxed{V = \sqrt{r g \tan \theta}}.
\]