5.13 Kinematics questions for IB Maths AI

51 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: Applications and Interpretation
Part of 5 Calculus
HL only

Questions by paper and level

Paper	Questions
Paper 1	22
Paper 2	28
Paper 3	1
Every paper	51

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 · AI-00048Paper 1 · 6 marks

A remote-controlled toy boat moves on a pond. Its velocity, in m s $^{- 1}$ , at time $t$ seconds is

v = (\begin{array}{c} 2 t \\ 12 - 3 t \end{array}), 0 \leq t \leq 6.

At $t = 0$ the boat is at the point with position vector $(\begin{array}{c} 5 \\ 0 \end{array})$ m.

(a)
Write down the acceleration of the boat.
1 mark
(b)
Find an expression for the position vector of the boat at time $t$ .
3 marks
(c)
Find the position of the boat at the instant it is moving parallel to the $x$ -axis.
2 marks

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Question 2 · AI-00421Paper 2 · 18 marks

At a festival a firework rocket is launched from a platform 12 metres above level ground. Let the origin be the point on the ground directly below the launch point, with the $x$ -axis horizontal and the $y$ -axis vertical. Distances are in metres and time, $t$ , in seconds after launch.

The rocket is launched with a speed of 30 m s $^{- 1}$ at an angle of 60° above the horizontal. After launch, its acceleration is modelled by $a = (\begin{array}{c} 0 \\ - 9.8 \end{array})$ m s $^{- 2}$ .

(a)
Write down the initial velocity of the rocket as a column vector, giving its components exactly.
2 marks
(b)
Find an expression for the velocity of the rocket at time $t$ .
2 marks
(c)
Show that the position vector of the rocket at time $t$ is $r = (\begin{array}{c} 15 t \\ 12 + 15 \sqrt{3} t - 4.9 t^{2} \end{array})$ .
3 marks
(d)
The rocket is designed to explode at its highest point. Find
(i) the time at which it explodes;
(ii) its height above the ground when it explodes;
(iii) the horizontal distance from the origin to the point where it explodes.
5 marks
(e)
On one occasion the rocket fails to explode and falls to the ground. Find the speed of the rocket when it hits the ground.
4 marks
(f)
Find the equation of the path of the rocket in the form $y = f (x)$ .
2 marks

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Question 3 · AI-01645Paper 3 · 29 marks

This question uses calculus to design how a tram accelerates and brakes between two stops.

A tram travels in a straight line between two stops, starting and finishing at rest. Its velocity, $v$ m s⁻¹, is recorded every 10 seconds during one journey of 80 seconds.

$t$ (s)	0	10	20	30	40	50	60	70	80
$v$ (m s⁻¹)	0	1.5	4.7	9.1	12.6	15.1	14.2	9.9	0

(a)
Use the trapezoidal rule with all the values in the table to estimate the distance between the two stops.
3 marks
(b)(i)
The tram company's engineer, Mireille, models the velocity by
$v (t) = \frac{t^{2} (80 - t)}{5000}, 0 \leq t \leq 80.$
Find the distance between the stops according to this model.
2 marks
(b)(ii)
Find the percentage difference between your answer to part (a) and your answer to part (b)(i), taking the value in part (b)(i) as the true value.
2 marks
(c)
Use differentiation to find the maximum velocity of the tram according to Mireille's model.
3 marks
(d)
Find the maximum acceleration of the tram according to the model, and the time at which it occurs.
3 marks
(e)
For passenger comfort, the magnitude of the acceleration of a tram must never exceed 1.3 m s⁻².
Show that Mireille's model satisfies this condition.
2 marks
(f)
Find the time at which the tram has travelled half of the distance between the stops.
3 marks
(g)(i)
On a new route, two stops are 900 m apart. Mireille uses a model of the same form,
$v (t) = c t^{2} (T - t), 0 \leq t \leq T,$
where $T$ seconds is the journey time and $c$ is a positive constant.
Show that $c = \frac{10 800}{T^{4}}$ .
3 marks
(g)(ii)
Show that the greatest magnitude of the acceleration during the journey is $\frac{10 800}{T^{2}}$ .
3 marks
(g)(iii)
Hence find the shortest journey time on the new route that satisfies the comfort condition in part (e).
2 marks
(g)(iv)
Find the maximum velocity of the tram on this journey.
2 marks
(h)
State one reason why Mireille's model may not be suitable for the new route.
1 mark

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Question 4 · AI-00049Paper 1 · 8 marks

Kofi kicks a football from ground level with a speed of 22 m s $^{- 1}$ at an angle of 35° above the horizontal. Taking the point where the ball is kicked as the origin, the velocity of the ball $t$ seconds after it is kicked is modelled by

v = (\begin{array}{c} 22 \cos 35 ° \\ 22 \sin 35 ° - 9.8 t \end{array}) {m s}^{- 1} .

(a)
Find the time taken for the ball to reach its maximum height.
2 marks
(b)
Find the maximum height reached by the ball.
3 marks
(c)
Find the horizontal distance travelled by the ball before it first hits the ground.
3 marks

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Question 5 · AI-00422Paper 2 · 20 marks

A car is tested on an elliptical track. Its position vector, in metres, relative to the centre of the track, $t$ seconds after the test begins is

r = (\begin{array}{c} 400 \cos (0.05 t) \\ 250 \sin (0.05 t) \end{array}) .

(a)
Write down the coordinates of the car's starting position.
1 mark
(b)
Find the velocity vector of the car at time $t$ .
2 marks
(c)
Find the speed of the car when $t = 10$ .
2 marks
(d)(i)
Show that the acceleration of the car is $a = - 0.0025 r$ .
3 marks
(d)(ii)
Hence state the direction of the acceleration.
1 mark
(e)
Find the time taken for the car to complete one lap of the track.
2 marks
(f)
Show that the square of the speed of the car is $400 \sin^{2} (0.05 t) + 156.25 \cos^{2} (0.05 t)$ .
2 marks
(g)
Find the maximum speed of the car and the coordinates of the points on the track where it occurs.
3 marks
(h)
Find the total time during one lap for which the speed of the car is greater than 15 m s $^{- 1}$ .
4 marks

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Question 6 · AI-00050Paper 1 · 6 marks

A car on a circular fairground ride moves so that its position vector relative to the centre of the ride, in metres, at time $t$ seconds is

r = (\begin{array}{c} 5 \cos (0.4 t) \\ 5 \sin (0.4 t) \end{array}) .

(a)
Find the velocity vector of the car at time $t$ .
2 marks
(b)
Show that the car moves with a constant speed of 2 m s $^{- 1}$ .
2 marks
(c)
Show that the acceleration of the car is always directed towards the centre of the ride.
2 marks

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