$A$ railway track is banked for a speed $v$ by elevating the outer rail by a height $h$ above the inner rail. If the distance between the two rails is $d$,then the radius of curvature of the track is ($g$ = gravitational acceleration).

Roads are banked on curves so that

$A$ 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 $1.5 \ m$)

Define optimum speed.

$A$ lorry is moving on a smooth circular path of radius $50 \ m$ with a velocity of $20 \ ms^{-1}$. Then the banking angle of the road is (Acceleration due to gravity $= 10 \ ms^{-2}$)

$A$ 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 maximum speed of the car is ............ $m/s$ $[g = 10\,m/s^2]$.