A particle moves along a straight horizontal line. Its velocity, $v\text{ ms}^{-1}$, is given by $v(t) = 2t - 2$ for $0 \le t \le 4$. State the time when the particle comes instantaneously to rest, and calculate the total distance travelled by the particle in the first 4 seconds.

1. Find when $v = 0$: $2t - 2 = 0 \implies \mathbf{t = 1\text{ s}}$. 2. Recognise the difference: displacement is $\int v \,dt$; total distance is $\int |v| \,dt$. Since the particle stops at $t = 1$ it changes direction. 3. Split : $\text{Distance} = \left|\int_0^1 (2t - 2)\,dt\right| + \int_1^4 (2t - 2)\,dt$. 4. Integrate : $\int (2t - 2)\,dt = t^2 - 2t$. 5. First segment ($0$ to $1$): $(1 - 2) - 0 = -1$, so distance $= |-1| = 1\text{ m}$. 6. Second segment ($1$ to $4$): $(16 - 8) - (1 - 2) = 8 - (-1) = 9\text{ m}$. Total $= 1 + 9 = \mathbf{10\text{ m}}$.

A particle has acceleration $a\text{ ms}^{-2}$ defined by $a(t) = \frac{1}{t^2} + \sin t$ for $t \ge 1$. Given that $v(1) = 1\text{ ms}^{-1}$, find a specific expression for $v(t)$.

1. Recall : $v(t) = \int a(t) \,dt$. 2. Rewrite with negative indices: $\int (t^{-2} + \sin t) \,dt$. 3. Integrate : $\frac{t^{-1}}{-1} - \cos t + C = -\frac{1}{t} - \cos t + C$. 4. State the general form: $v(t) = -\frac{1}{t} - \cos t + C$. 5. Apply the boundary condition $v(1) = 1$: $1 = -1 - \cos 1 + C$. 6. Solve : $C = 2 + \cos 1$. Specific expression: $\mathbf{v(t) = -\frac{1}{t} - \cos t + 2 + \cos 1}$.

A particle is found to have a velocity, $v\text{ ms}^{-1}$ , that can be expressed by the function: \[v(t) = t^3 \cos t, \quad t \ge 0\] Find an expression for the acceleration, $a(t)$ , of the particle. Hence, evaluate the exact acceleration of the particle at time $t = \pi$ .

Use the product rule: $u = t^3 \implies u' = 3t^2$ , $v = \cos t \implies v' = -\sin t$ . (M1) $a(t) = v'(t) = u'v + uv' = 3t^2\cos t - t^3\sin t$ . A1A1 Substitute $t = \pi$ : (M1) $a(\pi) = 3(\pi)^2\cos(\pi) - (\pi)^3\sin(\pi)$ . Since $\cos \pi = -1$ and $\sin \pi = 0$ : $a(\pi) = 3\pi^2(-1) - 0 = -3\pi^2\text{ ms}^{-2}$ . A1

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IB Maths AA SL · Unit 5: Calculus

IB Maths AA SL Kinematics Questions

Updated October 2026

Exam-style IB Maths AA SL kinematics questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step, showing the working that earns the marks — then try the practice questions and check your working against the mark scheme.

34 questions
Paper 1: 19
Paper 2: 15
4 easy
12 medium
15 hard
3 very hard
3 worked examples

Practise Kinematics questions →

🖩 How do I do this on my calculator? Keystrokes for the Casio fx-CG50, TI-84 Plus CE and TI-Nspire.

What you need to know

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. Kinematics — displacement, velocity, acceleration overview →

What's examined in AA SL kinematics

The question bank covers these kinematics question types (number of questions in brackets):

Instantaneous Kinematic Values (18)
Displacement and Total Distance (9)
Motion Analysis & Graphs (7)

Where students lose marks on kinematics

Examiner's noteKinematics: speed vs velocity, change of direction, distance vs displacement

Where marks are lost: Seen in every session. Candidates gave the maximum velocity when the maximum speed was asked, found maximum speed or acceleration by setting a derivative to zero instead of reading the graph and endpoints, did not link a change of direction to v = 0, used the GDC in degrees, and integrated v (displacement) when distance was asked.

Examiners flagged this in 4 exam series (May 2024, November 2024, May 2025, November 2025). Summarised in our own words from IB subject reports, IB Programme Resource Centre. Read the full note →

Examiner's noteGDC in the wrong angle mode, and degrees in radian formulas

Where marks are lost: Reported in four of the five sessions: kinematics and trigonometric models worked in degree mode, the sector-area formula used with the angle in degrees, the GDC's simultaneous-equation solver used in the wrong mode, and a bearing question done in radians.

Examiners flagged this in 3 exam series (May 2024, May 2025, November 2025). Summarised in our own words from IB subject reports, IB Programme Resource Centre. Read the full note →

Kinematics worked examples

Worked example 1: Displacement versus total distance · easy

A particle moves along a straight horizontal line. Its velocity, $v {ms}^{- 1}$ , is given by $v (t) = 2 t - 2$ for $0 \leq t \leq 4$ . State the time when the particle comes instantaneously to rest, and calculate the total distance travelled by the particle in the first 4 seconds.

Solution

6 steps

Examiner tip: For total distance, find every root of

v (t)

in the domain and split the integral there, or use

\int | v (t) | d t

on the GDC directly.

Worked example 2: Recovering velocity from acceleration · medium

A particle has acceleration $a {ms}^{- 2}$ defined by $a (t) = \frac{1}{t^{2}} + \sin t$ for $t \geq 1$ . Given that $v (1) = 1 {ms}^{- 1}$ , find a specific expression for $v (t)$ .

Solution

6 steps

Examiner tip: Do NOT evaluate trigonometric constants like

\cos 1

into decimals unless instructed — leave them exact. Remember the argument is

1

radian, not

1

degree.

Worked example 3: Kinematics via graphing technology · hard

A particle moves so that its displacement from a fixed point $P$ , after $t$ seconds, is $s (t) = 15 - 6 e^{0.8 t - 0.25 t^{2}}$ . Use your GDC to find the value of $t$ when the particle first returns to $P$ , and find its initial velocity at $t = 0$ .

Solution

6 steps

Examiner tip: In Paper 2, whenever the derivative is needed at a specific number, use the

\frac{d}{d x}

button. Hand-differentiating chained exponentials wastes time and is error-prone.

Try these IB Maths AA SL kinematics questions

Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area. One of them has its full worked solution here, open to everyone.

Question 1 · easy · 4 marks · Paper 2

A cricket ball is projected directly upwards from ground level. The vertical height of the cricket ball, $h$ metres above the ground, after $t$ seconds is modelled by the function: $h (t) = 13 t - 4.9 t^{2}, t \geq 0$

Use your graphic display calculator to find the times at which the cricket ball is exactly $3 m$ above the ground.
Calculate the maximum height reached by the cricket ball.

Attempt it and see the mark scheme →

Question 2 · medium · 5 marks · Paper 1

A particle is found to have a velocity, $v {ms}^{- 1}$ , that can be expressed by the function: $v (t) = t^{3} \cos t, t \geq 0$

Find an expression for the acceleration, $a (t)$ , of the particle.
Hence, evaluate the exact acceleration of the particle at time $t = π$ .

Full worked solution Free, no sign-in

Use the product rule: $u = t^{3} ⟹ u^{'} = 3 t^{2}$ , $v = \cos t ⟹ v^{'} = - \sin t$ . (M1)
$a (t) = v^{'} (t) = u^{'} v + u v^{'} = 3 t^{2} \cos t - t^{3} \sin t$ . A1A1
Substitute $t = π$ : (M1)
$a (π) = 3 (π)^{2} \cos (π) - (π)^{3} \sin (π)$ .
Since $\cos π = - 1$ and $\sin π = 0$ :
$a (π) = 3 π^{2} (- 1) - 0 = - 3 π^{2} {ms}^{- 2}$ . A1

Every question here has a full worked solution. See them all →

Question 3 · hard · 5 marks · Paper 1

A particle, moving in a straight line, is found to have a velocity $v (t) = \sin t + \cos (2 t)$ where $v$ is measured in ${ms}^{- 1}$ and $0 \leq t \leq 2 π$ . By using a suitable double angle trigonometric identity, analytically find the exact times $t$ when the particle is instantaneously at rest.

Attempt it and see the mark scheme →

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Revise kinematics for the exam

FAQ

How many IB Maths AA SL kinematics questions are there?

There are 34 exam-style kinematics questions in the AA SL question bank (Paper 1: 19 · Paper 2: 15), graded 4 easy, 12 medium, 15 hard, 3 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is kinematics on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 19 · Paper 2: 15. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.

Where can I get the mark schemes?

Open the AA 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.

What else is free here

Save your answers and track your progressA free account remembers the questions you have done on every topic and brings you back to this page.

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Teaching this topic?

Free ready-to-teach lesson (slides and teacher notes): Kinematics: displacement, velocity, acceleration.

IB Maths AA SL Kinematics Questions

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What you need to know

What's examined in AA SL kinematics

Where students lose marks on kinematics

Kinematics worked examples

Worked example 1: Displacement versus total distance · easy

Solution

Worked example 2: Recovering velocity from acceleration · medium

Solution

Worked example 3: Kinematics via graphing technology · hard

Solution

Try these IB Maths AA SL kinematics questions

Question 1 · easy · 4 marks · Paper 2

Question 2 · medium · 5 marks · Paper 1

Full worked solution Free, no sign-in

Question 3 · hard · 5 marks · Paper 1

Revise kinematics for the exam

FAQ

What else is free here

Teaching this topic?

More AA SL Unit 5 topics