12.3: 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 ut. 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 un.
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 un. 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, ut undergoes changes at each instant, indicating the changes in the direction of un.
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.
