Vector Calculus — Position, Velocity & Acceleration
Differentiating and integrating position vectors to find velocity and acceleration — practice questions for VCE Specialist Mathematics Units 3&4.
What's inside
- A short recap of the method before the questions start
- Practice questions ordered from easiest to hardest
- Space to show full working, plus an answer key
Sample questions
Question 1
A particle's position vector at time t seconds is r(t) = (t² − 3t)i + (2t + 1)j metres. Find its velocity vector at t = 2.
Show worked solution
Velocity is the derivative of position: v(t) = r′(t) = (2t − 3)i + 2j. Substituting t = 2 gives v(2) = (2(2) − 3)i + 2j = i + 2j m/s.
Question 2
A particle has velocity vector v(t) = 4t i − 3j m/s and initial position r(0) = 2i + j m. Find its position vector at t = 3 seconds.
Show worked solution
Integrate each component: r(t) = ∫4t dt · i + ∫−3 dt · j = (2t² + C₁)i + (−3t + C₂)j. Using r(0) = 2i + j gives C₁ = 2 and C₂ = 1, so r(t) = (2t² + 2)i + (1 − 3t)j. At t = 3: r(3) = (2(9) + 2)i + (1 − 9)j = 20i − 8j m.
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