$A$ 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).

$A$ road is $10 \text{ m}$ wide. Its radius of curvature is $50 \text{ m}$. The outer edge is above the inner edge by a distance of $1.5 \text{ m}$. This road is most suited for the velocity $[g = 9.8 \text{ m/s}^2]$: (in $\text{ m/s}$)

$A$ 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 $1.5 \ m$)

$A$ train has to negotiate a curve of radius $r \ m$. The distance between the rails is $\ell \ m$ and the outer rail is raised above the inner rail by a distance of $h \ m$. If the angle of banking is small,the safety speed limit on this banked track is:

What is optimum speed? Write its equation.

$A$ 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 ....... $m\,s^{-1}$.