Short on time? This page is the recap. For the explanations, the hovercraft diagram and the worked examples, use the full study guide. BC-only material. Prerequisites (vector derivatives 9.4, vector integrals 9.5, arc length 9.3, line motion 4.2 and 8.2) are on the Calculus BC roadmap.
Recap
- Differentiate position to get velocity and acceleration. Integrate velocity to get displacement and position.
- Speed is the length of the velocity vector. It is never negative.
- Signs of x′ and y′ give the direction: right/left and up/down.
- At rest only when x′ = 0 and y′ = 0 at the same time.
- Distance integrates speed; displacement integrates velocity. Distance ≥ length of displacement.
Key relationships
| Quantity | Formula | Type |
|---|---|---|
| Velocity | v(t) = ⟨x′(t), y′(t)⟩ | vector |
| Acceleration | a(t) = ⟨x″(t), y″(t)⟩ | vector |
| Speed | √((x′)² + (y′)²) | number |
| Speed increasing? | sign of x′x″ + y′y″ (positive: increasing) | test |
| Displacement on [a, b] | ∫ from a to b of v(t) dt = r(b) − r(a) | vector |
| Position | r(b) = r(a) + ∫ from a to b of v(t) dt | point |
| Total distance on [a, b] | ∫ from a to b of √((x′)² + (y′)²) dt | number |
| Average speed | distance ÷ time | number |
Assumptions
- Radians in every trigonometric velocity component.
- Calculator values: keep 4 or more decimal places, round to 3 at the end.
- Units: if position is in metres and t in seconds, speed is in m/s and distance in m.
Mistakes to avoid
- Speed = x′ + y′. It is √((x′)² + (y′)²).
- “At rest” from one zero component. Check both.
- ∫ v dt as distance. That is displacement; distance needs the speed.
- Forgetting r(a) when asked for a position.
- Judging speeding up from the size of a(t). Use the sign of x′x″ + y′y″.
- Degree mode on the calculator.
Quick self-check
- r(t) = ⟨t², 3t − t³⟩. Find the speed at t = 2. (v(2) = ⟨4, −9⟩, speed √97 ≈ 9.849)
- A particle has constant velocity ⟨4, 3⟩ for 0 ≤ t ≤ 2. How far does it travel? (speed 5, distance 10)
- v(t) = ⟨cos t, −sin t⟩ for 0 ≤ t ≤ π. Find the displacement and the distance. (displacement ⟨0, −2⟩; speed is 1, so distance π)
Next: practice questions.