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WASpecialist MathematicsSyllabus dot point

How do we describe the position of a moving particle by a vector function and differentiate it to get velocity and acceleration?

Use vector functions of time and differentiate them to find velocity, speed and acceleration along a path

WACE Specialist Unit 3 vector calculus: position vectors as functions of time, componentwise differentiation to velocity and acceleration, speed as the magnitude of velocity, the cartesian path equation, and motion problems, with a worked example.

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  1. What this dot point is asking
  2. Position as a vector function of time
  3. Differentiation is componentwise
  4. Speed is a scalar
  5. Using the calculus on motion

What this dot point is asking

SCSA wants you to model motion with a vector function, differentiate it to obtain velocity and acceleration, find speed, and describe the path.

Position as a vector function of time

A path is described by r(t)=x(t) i+y(t) j+z(t) k\mathbf{r}(t) = x(t)\,\mathbf{i} + y(t)\,\mathbf{j} + z(t)\,\mathbf{k}, giving the particle's position at each time tt. Evaluating at particular times gives points on the path; eliminating the parameter tt between the component equations gives the cartesian relation of the curve.

Differentiation is componentwise

Each component is differentiated separately using ordinary calculus. The velocity vector is tangent to the path, pointing in the direction of motion; the acceleration vector records how velocity changes.

Speed is a scalar

Using the calculus on motion

To find when a particle is momentarily at rest, set v(t)=0\mathbf{v}(t) = \mathbf{0} (all components zero). To find when velocity is perpendicular to acceleration, set v⋅a=0\mathbf{v}\cdot\mathbf{a} = 0. Position, displacement and distance follow by evaluating or integrating as needed.