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SL AA · Calculus
Kinematics — displacement, velocity, acceleration (IB Math AA SL)
Updated October 2026
Quick answer: Velocity is the derivative of displacement and acceleration is the derivative of velocity. The particle is at rest when v = 0, displacement is ∫v dt, and distance travelled is ∫|v| dt.
s(t), v(t) = s'(t), a(t) = v'(t). SL AA Paper 2 asks 'when does the particle come to rest?' and expects you to spot v = 0.
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v = ds/dt and a = dv/dt. Going back, s = ∫v dt: use the initial position to find the constant.
At rest means v = 0. The particle changes direction only where v changes sign, so check the sign on each side.
Displacement from t₁ to t₂ is ∫v dt; total distance is ∫|v| dt. They differ whenever the particle turns back.
The particle is speeding up when v and a have the same sign, and slowing down when they have opposite signs.
On Paper 2, the GDC can integrate |v| directly; on Paper 1, split the integral at each zero of v.
Worked example
A particle moves in a straight line with velocity v(t) = t² − 6t + 8 m s⁻¹ for 0 ≤ t ≤ 5. Find when it is at rest, its acceleration at t = 1, its displacement over the 5 seconds and the total distance it travels.
v = (t − 2)(t − 4) = 0, so the particle is at rest at t = 2 and t = 4 seconds.
A calculator is allowed on AA Paper 2, not on Paper 1. Keystrokes for the TI-84 Plus CE, TI-Nspire CX II and Casio fx-CG50, with a checked worked example:
What is the difference between displacement and distance in kinematics?
Displacement is ∫v dt, where motion backwards counts as negative. Distance is ∫|v| dt, which counts every metre travelled. They are equal only if the particle never changes direction.
Where can I practise kinematics questions?
On the Kinematics questions page: 34 exam-style SL AA questions with full mark schemes, plus free step-by-step worked examples.
Can I use my calculator for kinematics?
In SL AA, Paper 1 is non-calculator and the other papers allow a GDC, so learn the exact method as well as the calculator one. The Numerical derivative and integral guide gives the keystrokes for the TI-84 Plus CE, TI-Nspire CX II and Casio fx-CG50.