On a railway curve,the outside rail is laid higher than the inside one so that the resultant force exerted on the wheels of the rail car by the tops of the rails will:

$A$ 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 $1.5 \ m$)

The safe speed of a vehicle over a smooth banked road of radius $150\ m$ is $10\ m/s$. If the width of the road is $7.5\ m$,the height of the outer edge is: (in $m$)

What is the speed of a vehicle moving on a flat and smooth (frictionless) circular path of radius $r$?

Define optimum speed.

What is optimum speed? Write its equation.