Short on time? This page is the recap. For explanations, diagrams and worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Recap
- Fix an origin and a positive direction. Position x(t) can be negative.
- Velocity v(t) = x′(t). Acceleration a(t) = v′(t) = x″(t). Speed = |v(t)|.
- v > 0: moving in the positive direction. v < 0: moving in the negative direction. v = 0: at rest.
- The object changes direction only where v changes sign.
- On a velocity graph, the slope is the acceleration and the distance from the t-axis is the speed.
Key relationships
| Signs | Result |
|---|---|
| v and a both positive, or both negative | speeding up |
| v and a with opposite signs | slowing down |
| a = 0 on an interval | constant velocity |
| Quantity | Formula | Units (m, s) |
|---|---|---|
| Average velocity on [t₁, t₂] | (x(t₂) − x(t₁))/(t₂ − t₁) | m/s |
| Average acceleration on [t₁, t₂] | (v(t₂) − v(t₁))/(t₂ − t₁) | m/s² |
| Displacement | x(t₂) − x(t₁) | m |
| Total distance | sum of |change in x| between turning points | m |
Assumptions behind the method
- Position is differentiable, so velocity exists. Where a velocity graph has a corner, a(t) does not exist.
- To find total distance from positions, you must know every time where v changes sign in the interval.
- “Up” or “right” is positive only if the problem says so. Check the convention.
Mistakes to avoid
- “Negative acceleration means slowing down.” Compare signs with v first.
- Taking v = 0 as a change of direction without checking the sign on each side.
- Using a = 0 to find turning points. Turning points come from v.
- Giving speed as a negative number.
- Using |final − initial| as total distance when the object turns.
- Wrong units: acceleration is m/s², not m/s.
Quick self-check
- x(t) = t² − 6t + 5. Find v(1), a(1) and the speed at t = 1. Is the particle speeding up? (v(1) = −4, a = 2, speed 4. Opposite signs, so slowing down.)
- When is that particle at rest? Does it change direction then? (t = 3. Yes: v = 2t − 6 changes from negative to positive.)
- A stone falls with up as positive: v < 0 and a < 0. Speeding up or slowing down? (Speeding up: same signs.)
Next: practice questions.