Trapezoidal speed time curve of Main line service

The speed-time curve of a main line service is best and most easily replaced by a trapezoid. Since the area of the speed- time curve represents the total distance travelled hence the areas of the two curves should be same.

Trapezoidal speed–time curve is close  approximation of main line service where the duration of free running time is long.

Let, D = distance between the stops in km, T = actual running time of train in second, α = acceleration in km/h/sec, β = retardation in km/h/sec, Vm = maximum or the crest speed of train in km/h, and Va=  average speed of train in km/h.

Area under the trapezoidal speed–time curve gives the total distance between the two stops (D). ∴ The distance between the stops (D) = area of Trapezium OABCO

Quadrilateral Speed Time Curve of Urban & Sub urban Service

The speed-time curve of an urban service can be replaced by an equivalent speed-time curve of simple quadrilateral shape.