Motion: Classical Mechanics and Calculus

The Study of Motion: Classical Mechanics

This course introduces classical mechanics, a branch of physics that describes the motion of objects. We will focus on mathematically describing motion.

In a previous lecture, vector quantities for position, velocity, and acceleration in 3D were introduced:

  • Position vector: \vec{r}(t) =
  • Velocity vector: \vec{v}(t) =
  • Acceleration vector: \vec{a}(t) =

To develop physical models from these, calculus methods are essential for relating position, velocity, and acceleration.

1.1 Motion in 1D, 2D and 3D

Motion in 1D

Motion in one dimension (1D) simplifies problems significantly, making it ideal for introducing physics concepts. If an object moves only in the y-dimension (no x or z motion), its position vector changes only in the y-component:

  • \vec{r}(t) =
  • The x and z components can generally be any constant (not necessarily zero).
  • For instance, if z=0, motion is in the x-y plane. If x=j (a constant), 1D motion parallel to the y-axis occurs along that line (Figure 1).

In 1D motion, x and z components of velocity and acceleration are zero:

  • Velocity vector: \vec{v}(t) =

  • Acceleration vector: \vec{a}(t) =

  • The magnitude of the velocity vector, |\vec{v}|, is the speed (v_s = |\vec{v}| ), which describes how fast an object is moving without considering direction.

  • When motion is strictly 1D, we can simplify by working only with scalar quantities like y, vy, and ay, avoiding vector notation.

  • More generally, 1D motion means an object moves along a straight line in 3D space, where all its points in time lie on the same line (Figure 2).

  • To simplify, we always align our coordinate axes parallel to the problem's motion. We can choose an origin (P(0,0,0) = O) and align positive/negative axes to suit our needs.

  • Often in 1D, we set up coordinates so the object starts at the origin and moves only in the x-direction, with y and z components always zero (Figure 3).

Motion in 2D and 3D

  • In reality, objects often move in all three dimensions. We start with 1D to grasp basics before moving to complexity.
  • Our