The radius of curvature of a railway track at a bend is 500 m. The distance between the two tracks is 1 m and the outer line is 4 cm higher than the inner. At what maximum speed can a train bend round the curve without exerting lateral pressure on the outer line?

Explanation

If the speed of the train is v, radius of curvature of the tracks is r and the angle of banking is `theta`, then
tan `theta = (v^(2))/(rg)`
If the distance between the two lines is x and the difference in their heights is h, then
tan`theta = (h)/(x)`
`therefore " " (v^(2))/(r g) = (h)/(x) " " or, " " v^(2) = (h r g )/(x) " " or, " " v = sqrt((h r g)/(x))`
`therefore " " v = sqrt((0.04 xx 500 xx 9.8)/(1)) = 14 m cdot s^(-1) = 50.4 "km" cdot h^(-1)`.