Curvilinear Motion: Normal and Tangential Components

When a car traverses a curved road, its motion can be elucidated by breaking it down into tangential and normal components. The car-centric coordinates attached to the vehicle move with it.

The positive direction of the t-axis aligns with the increasing position of the car along the curved path, denoted by the unit vector u_t. Simultaneously, the n-axis, perpendicular to the t-axis, dissects the curved path into differential arc segments, each forming the arc of a circle with a radius of curvature and a center of curvature. The positive direction of the n-axis points toward the center of curvature, designated by the unit vector u_n.

On the other hand, the normal component is related to the curvature of the road. The n-axis, perpendicular to the t-axis, is involved in dissecting the curved path into tiny differential arc segments. Each of these segments forms the arc of a circle with a specific radius of curvature and a center of curvature. The positive direction of the n-axis points towards the center of this curvature and is defined by the unit vector u_n. This component helps describe how the car is deviating from a straight-line path due to the curvature of the road.

The car's velocity remains tangent to the road, possessing only a t-component. By differentiating the velocity expression with respect to time, the car's acceleration is derived. Importantly, u_t undergoes changes at each instant, indicating the changes in the direction of u_n.

In essence, for curvilinear motion, the car's acceleration manifests in both tangential and normal components, providing a comprehensive understanding of how the vehicle navigates along the curved trajectory.