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**Hint:**When roads are slanting, they are said to be blanked. While taking a turn, banking of roads is needed to increase safety. At the optimum speed the friction between car tyres and surface of the road is negligible. The optimum speed = \[\sqrt {rg\operatorname{Tan} \theta } \] where ‘r’ is the radius of the track and\[\theta \] is the angle of banking.

**Complete answer:**

Optimum speed is the speed with which one can take turns on banked roads without wear and tear.

\[{v_o} = \sqrt {rg\tan \theta } \], where \[{v_o}\] = optimum speed, r= radius, g= gravitational constant. Roads are most often banked for the average speed of vehicles passing over the proper velocity or optimum ‘v’ on a road banked by an angle \[\theta \] with the horizontal is given by= \[\sqrt {Rg\tan \theta } \] where R is the radius of the curvature of the road, g is the acceleration due to gravity. Hence, the right answer is: \[\sqrt {Rg\tan \theta } \].

**Note:**When a vehicle is moving along a curved path it requires centripetal forces along the curve so that it does not tend to skid over the road. Maximum optimum speed depends on: Radius of the curved path; coefficient of the friction; angle on inclination. We see the angle of banking is independent of the mass of the vehicles.

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