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MathidiaMathematics Academy
Mathematical mechanics
Lesson 61 of 75
Introduction

Vector kinematics

Objective. Differentiate vector position to analyse motion.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Vector kinematics describes motion using a position vector r(t) that depends on time, with velocity v(t) = dr/dt and acceleration a(t) = dv/dt found by differentiating each component separately. Given r(t) = (t², 3t), velocity is v(t) = (2t, 3) and acceleration is a(t) = (2, 0), showing constant acceleration in the x-direction only. Speed is the magnitude |v(t)|, while direction is given by the unit vector v/|v|. This component-wise approach extends naturally to three dimensions for projectile or orbital motion.

Worked example

A particle has position vector r(t) = (t³, 2t) for vector kinematics. Find its acceleration vector at t = 2.

2 … 2
Illustrated card showing 2 … 2
  1. 1Velocity: v(t) = (3t², 2)
  2. 2Acceleration: a(t) = (6t, 0)
  3. 3Substitute t=2: a(2) = (12, 0)
  4. 4State final vector: a(2) = (12, 0)

Answer: (12, 0)

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Common mistake. Computing speed by adding the magnitudes of velocity components instead of using Pythagoras; for v=(3,4), speed is √(3²+4²)=5, not 3+4=7.

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