12.1: Curvilinear Motion: Rectangular Components
Curvilinear motion characterizes the movement of a particle or object along a curved path, notably evident when envisioning a car navigating a winding road. If the car starts at point A, its position vector is established within a fixed frame of reference, where the ratio of the position vector to its magnitude signifies the unit vector pointing in the position vector's direction.
As the car advances, its position evolves over time. Quantifying the car's velocity involves computing the time derivative of the position vector. Notably, in the fixed frame of reference, the direction of unit vectors remains constant over time.
The velocity vector, expressing the car's speed and direction, can be dissected into rectangular components. Dividing this vector by its magnitude unveils the unit vector along the direction of the velocity of the car, akin to the car's heading on the curved road.
Further, taking the time derivative of the velocity vector gives the acceleration vector, representing how the car's speed or direction changes with time. Normalizing this acceleration vector by its magnitude yields the unit vector for the acceleration of the car, disclosing the direction of the car's acceleration. In essence, these principles provide a conceptual framework for understanding the intricacies of a car engaged in curvilinear motion.
