The radius of a curved road on a national highway is $R$. The width of the road is $b$. The outer edge of the road is raised by $h$ with respect to the inner edge so that a car with velocity $v$ can pass safely over it. The value of $h$ is

$A$ car of mass $1000 \, kg$ negotiates a banked curve of radius $90 \, m$ on a frictionless road. If the banking angle is $45^\circ$,the speed of the car is ......... $m \, s^{-1}$.

$A$ car is moving with speed $v$ on a banked road of radius $50 \ m$,width $10 \ m$,and height $1.5 \ m$. The velocity $v$ of the car is ......... $m/s$.

$Assertion$ : There is a stage when frictional force is not needed at all to provide the necessary centripetal force on a banked road.
$Reason$ : On a banked road,due to its inclination the vehicle tends to remain inwards without any chances of skidding.

Does the motion of a vehicle on a level circular path depend on the mass of the vehicle?

$A$ railway line is taken round a circular arc of radius $1000 \ m$,and is banked by raising the outer rail $h \ m$ above the inner rail. If the lateral force on the inner rail when a train travels round the curve at $10 \ ms^{-1}$ is equal to the lateral force on the outer rail when the train's speed is $20 \ ms^{-1}$,then the value of $4g \tan \theta$ is equal to: (The distance between the rails is $1.5 \ m$)