Final answer:
The component differential equations of motion include calculations of a particle's position, velocity, and acceleration through integral calculus. These equations are essential for describing motion in classical mechanics and quantum mechanics, where the Schrödinger equation provides the wave function for a particle's location probability.
Step-by-step explanation:
The component differential equations of motion for a particle describe how the particle's position, velocity, and acceleration vary over time and can be solved to determine the motion of the particle in space. In classical mechanics, these equations often start with knowing the acceleration as a function of time, which leads to finding the velocity by integration, and then the position by integrating the velocity.
When analyzing motion in two or three dimensions, we break it down into its components along the x- and y-axes (and the z-axis for three-dimensional motion). By applying the principles of conservation of momentum for collisions or using kinematic equations derived from integral calculus, we can find the velocity and position at any time t.