Follow a path through its derivatives

Differentiate each component of a position vector to obtain velocity, then differentiate again for acceleration. Speed is the magnitude of velocity; it is a scalar and cannot be negative.

Integrate velocity component by component to recover position, using the initial position to determine the constant vector. Integrating speed gives distance travelled, whereas integrating velocity gives displacement.

\displaystyle \mathbf v=\mathbf r',\quad\mathbf a=\mathbf r'',\quad\text{speed}=|\mathbf v|

WORKED EXAMPLE

For r(t) = ( $t^{2}$ , 3t), find speed at t = 2.

Velocity is (2t, 3).
At t = 2 it is (4, 3).
Speed is √(16 + 9) = 5.

Assumed knowledge

Differentiate and integrate elementary functions from Mathematical Methods. Use vector components and magnitudes, initial conditions, and the distinction between displacement and distance.

Study Specialist Mathematics alongside, or after, Mathematical Methods. Later units build on earlier units.

QCAA prior-learning guidance ↗

Follow a path through its derivatives

For r(t) = (t2t^{2}t2, 3t), find speed at t = 2.

Put the idea to work.

For r(t) = ( $t^{2}$ , 3t), find speed at t = 2.