Velocity in Frictionless Banked Turn


A banked turn (aka. banking turn) is a turn or change of direction in which the vehicle banks or inclines, usually towards the inside of the turn. For a road or railroad this is usually due to the roadbed having a transverse down-slope towards the inside of the curve. The bank angle is the angle at which the vehicle is inclined about its longitudinal axis with respect to the horizontal.

As opposed to a vehicle riding along a flat circle, inclined edges add an additional force that keeps the vehicle in its path and prevents a car from being “dragged into” or “pushed out of” the circle (or a railroad wheel from moving sideways so as to nearly rub on the wheel flange). This force is the horizontal component of the vehicle’s normal force. In the absence of friction, the normal force is the only one acting on the vehicle in the direction of the center of the circle.

Solving for velocity we have the formula shown.

This provides the velocity that in the absence of friction and with a given angle of incline and radius of curvature, will ensure that the vehicle will remain in its designated path. The magnitude of this velocity is also known as the “rated speed” (or “balancing speed” for railroads”) of a turn or curve.Notice that the rated speed of the curve is the same for all massive objects, and a curve that is not inclined will have a rated speed of 0.

Related formulas


vvehicle velocity (m/s)
rradius of curvature (m)
gStandard gravity
θangle of incline (degrees)