12.1:
Curvilinear Motion: Rectangular Components
Curvilinear motion is a type of motion where a particle or an object moves along a curved path involving changes in direction.
If the object is at point A, its position vector is defined using a fixed frame of reference. Here, the ratio of the position vector with its magnitude represents the unit vector in the direction of the position vector.
As the object moves forward, its position changes with time. The velocity of the object is calculated by taking the time derivative of the position vector. Here, for the fixed frame of reference, the direction of unit vectors does not change with time.
Here, the velocity vector can be expressed using rectangular components, and dividing it with the magnitude of velocity vectors gives the unit vector along the direction of the velocity vector.
Taking the time derivative of the velocity vector gives the acceleration vector for the object in rectangular components.
Dividing the acceleration vector with its magnitude gives the unit vector along the direction of the acceleration vector.
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.