Prerequisites: particle approximation and context

Summary

Newton’s laws are commonly written using the “particle” (point-mass) approximation: bodies are treated as having negligible volume. This is valid when the internal structure and motion of parts can be ignored and when the distance between bodies is much larger than their sizes. For example, Earth and the Sun can be approximated as pointlike when studying the Earth’s orbit, even though Earth cannot be treated as pointlike for phenomena on its surface. The mathematical description of motion (kinematics) also underpins the prerequisites: positions are specified using coordinates that vary with time. In the simplest one-dimensional case, a body’s location is given by a single coordinate relative to an origin, and its average velocity over a time interval is defined from the change in position divided by the change in time. Calculus then allows the definition of instantaneous velocity as the limit of average velocity as the time interval shrinks to zero, and acceleration is defined similarly as the time derivative of velocity (equivalently, the second derivative of position).

Key Points

• 1

  Particle approximation treats bodies as pointlike when size and internal motion are negligible compared with the problem’s relevant distances and scales.

• 2

  Kinematics uses time-dependent coordinates to describe motion, with average velocity defined over intervals and instantaneous velocity obtained via limits (derivatives).

• 3

  Acceleration is defined as the time derivative of velocity, i.e., the second derivative of position with respect to time.