5.9 Kinematics questions for IB Maths AA

41 questions on Kinematics, written by Augment in the style of IB exam papers. Six of them are below in full, to answer on paper.

IB Mathematics: Analysis and Approaches
Part of 5 Calculus
SL and HL

Questions by paper and level

SL and HL questions are for students at either level. HL only questions cover what only HL students study.

Paper	SL and HL	HL only	All
Paper 1	14	2	16
Paper 2	18	7	25
Every paper	32	9	41

Counted on 10 November 2026

Questions to try

Written by Augment in the style of IB exams, not past papers. Answer them on paper, then check each one against its markscheme with a free account.

Question 1 · AA-00727Paper 1 · 6 marks

A particle P travels in a straight line. At time $t$ seconds, where $t \geq 0$ , P has velocity $v (t) = 6 t^{2} - 2 t + 1$ , in m s $^{- 1}$ .

The displacement of P from a fixed point O on the line is $s$ metres. When $t = 0$ , $s = 4$ .

(a)
Find an expression for $s$ in terms of $t$ .
4 marks
(b)
Find the displacement of P from O when $t = 3$ .
2 marks

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Question 2 · AA-00179Paper 2 · HL only · 7 marks

A particle P travels along a straight line. At time $t$ seconds, the displacement of P from a fixed point O is $s$ metres, where

s (t) = 4 \sin (t + \frac{π}{3}) - 2 \cos (t - \frac{π}{6}), for 0 \leq t \leq 5.

(a)
Show that $s (t) = \sin t + \sqrt{3} \cos t$ .
3 marks
(b)
Find the value of $t$ when P is first at rest.
2 marks
(c)
Find the total distance travelled by P.
2 marks

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Question 3 · AA-00751Paper 1 · 15 marks

A particle P moves along a straight line. At time $t$ seconds, its displacement from a fixed point O on the line is $s$ metres and its velocity is $v$ m s $^{- 1}$ , where

v (t) = 4 \cos 2 t - 2, 0 \leq t \leq π .

When $t = 0$ , P is $1$ metre from O, so that $s = 1$ .

(a)
Write down the initial velocity of P.
1 mark
(b)
Find the values of $t$ for which P is at rest.
4 marks
(c)
Find the acceleration of P when $t = \frac{π}{4}$ .
2 marks
(d)
Find an expression for $s$ in terms of $t$ .
4 marks
(e)
Find the total distance travelled by P during the first $\frac{π}{2}$ seconds. Give your answer in the form $a \sqrt{3} + \frac{π}{b}$ , where $a, b \in Z^{+}$ .
4 marks

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Question 4 · AA-00363Paper 2 · 14 marks

A particle P travels in a straight line. At time $t$ seconds, the velocity of P is $v$ m s $^{- 1}$ , where

v (t) = 4 \cos (1.5 t) + 0.5 t - 1, 0 \leq t \leq 6.

(a)
Write down the initial velocity of P.
1 mark
(b)
Find the values of $t$ at which P is at rest.
3 marks
(c)
Find the acceleration of P when it changes direction for the second time.
3 marks
(d)
Find the total distance travelled by P.
3 marks
(e)
Find the displacement of P from its initial position when $t = 6$ .
2 marks
(f)
Find the total length of time for which the velocity of P is negative.
2 marks

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Question 5 · AA-01133Paper 1 · HL only · 7 marks

Water drains from a tank through a hole in its base. The depth of water in the tank, $h$ metres, at time $t$ minutes is modelled by the differential equation

\frac{d h}{d t} = - 0.2 \sqrt{h} .

Initially the depth of water is 4 metres.

(a)
Show that $h = (2 - 0.1 t)^{2}$ , for $0 \leq t \leq 20$ .
4 marks
(b)
Find the time taken for the depth of water in the tank to fall to 1 metre.
2 marks
(c)
Write down the time taken for the tank to empty.
1 mark

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Question 6 · AA-00651Paper 2 · 5 marks

A cyclist rides along a straight road. Her distance, $s$ metres, from the start after $t$ seconds was recorded by a sensor. Some of the results are shown in the following table.

$t$ (s)	$1.9$	$1.99$	$2$	$2.01$	$2.1$
$s$ (m)	$13.87$	$14.8852$	$15$	$15.1152$	$16.17$

(a)
Find the average velocity of the cyclist between $t = 2$ and $t = 2.1$ .
2 marks
(b)(i)
Find the average velocity of the cyclist between $t = 1.99$ and $t = 2$ .
1 mark
(b)(ii)
Find the average velocity of the cyclist between $t = 2$ and $t = 2.01$ .
1 mark
(c)
Hence estimate the velocity of the cyclist at $t = 2$ .
1 mark

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