5 · IB Maths
Calculus of motion
Differentiate displacement to get velocity, and again for acceleration. Watch the particle shuttle along the line while the three graphs explain every hesitation, and see the tangent to s(t) whose gradient is v.
Predict first
Choose s(t) = t² − 4t + 3 in the Motion menu and set Time t to 4 s. How do the distance travelled and the displacement compare?
s(0) = 3 and s(4) = 3, so the displacement is 0. v = 2t − 4 = 0 at t = 2 s, where s = −1; the particle goes 4 m back and 4 m forward, so the distance is 8 m.
Check it in the model: Select s(t) = t² − 4t + 3, drag the Time, t slider to 4 s and read Distance travelled and Displacement.
s(t) v(t) a(t) · dashed line = tangent to s(t)
Try this
- Choose s(t) = t³ − 6t² + 9t, pause, and move t to 1 s and 3 s. What is v at each time, and what does the particle do there?
- Choose s(t) = t² − 4t + 3 and set t = 4 s. Compare the distance travelled with the displacement and explain the difference.
- Choose 4 sin t and find a time where the particle is speeding up and one where it is slowing down. How do the signs of v and a compare?
Exam tip: the particle changes direction where v = 0 and v changes sign, not where s = 0, and total distance is the integral of |v| over time, so split the integral at each turning time.
More on 5: Chain and product rules · Second derivative and points of inflexion · Trapezoidal rule and Riemann sums · Displacement and distance travelled · Euler's method and differential equations · Implicit differentiation · Related rates of change · Area under a curve · Limits · Maclaurin series · Newton–Raphson method (enrichment) · Optimisation · Slope fields · Tangents and the derivative · Volume of revolution