IB Maths AI SL · Unit 5: Calculus
Exam-style IB Maths AI SL kinematics questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.
The link between differentiation and integration in a physical context. SL AI expects you to know v = ds/dt and a = dv/dt, plus how to reverse the process by integrating. Kinematics — displacement, velocity, and acceleration overview →
$s(t) = 2t^3 - 5t^2 + 4t$ metres. Find $v(t)$ and state the exact initial velocity.
1. Rule: $v(t) = \tfrac{ds}{dt}$.
2. Differentiate: $v(t) = 6t^2 - 10t + 4$.
3. Initial: $t = 0$.
4. Evaluate: $v(0) = 4$.
5. State: $\mathbf{4\text{ m s}^{-1}}$.
$v(t) = t^2 - 7t + 10\text{ m s}^{-1}$ for $t \ge 0$. Find the exact times the object is at rest.
1. At rest: $v(t) = 0$.
2. Set up: $t^2 - 7t + 10 = 0$.
3. Factorise: $(t-2)(t-5) = 0$.
4. Solve: $t = 2$ or $t = 5$.
5. State: $\mathbf{t = 2\text{ s and } t = 5\text{ s}}$.
With $v(t) = t^2 - 7t + 10\text{ m s}^{-1}$, use the GDC to find the exact total distance travelled in the first $6$ seconds.
1. Rule: total distance $= \int_0^6 |v(t)|\,dt$.
2. Set up: $\int_0^6 |t^2 - 7t + 10|\,dt$.
3. GDC: wrap the function in Abs() inside the integral tool.
4. Execute: numeric evaluation.
5. State: $\mathbf{13.5\text{ m}}$.
Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area.
A particle moves along a straight line such that its velocity, \(v\) in \(\text{m s}^{-1}\), at time \(t\) seconds is given by \(v(t) = t^2 - 4t + 3\) for \(t \ge 0\).
The velocity \(v\) in \(\text{m s}^{-1}\) of a particle moving in a straight line after \(t\) seconds is given by \(v(t) = 3t^2 - 18t + 15\) for \(t \ge 0\).
The velocity \(v\) in \(\text{m s}^{-1}\) of a particle moving in a straight line after \(t\) seconds is given by \(v(t) = 3t^2 - 24t + 36\) for \(t \ge 0\).
There are 12 exam-style kinematics questions in the AI SL question bank (Paper 1: 2 · Paper 2: 10), graded 4 very hard, 6 hard, 2 medium. Every question has a full IB-style mark scheme (M, A and R marks).
Both. In the bank, Paper 1: 2 · Paper 2: 10. Practise with your GDC — AI papers expect calculator methods throughout.
Open the AI SL Unit 5 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.