AHL 5.13—Kinematic problems Questionbank

Practice AHL 5.13—Kinematic problems with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

SLPaper 2

Note: In this question, distance is in metres and time is in seconds.

A particle P moves in a straight line for five seconds. Its acceleration at time $t$ is given by $a = 3{t^2} - 14t + 8$ , for $0 \leqslant t \leqslant 5$ .

When $t = 0$ , the velocity of P is $3{\text{ m}}\,{{\text{s}}^{ - 1}}$ .

Write down the values of $t$ when $a = 0$ .

[2]

Hence or otherwise, find all possible values of $t$ for which the velocity of P is decreasing.

[2]

Find an expression for the velocity of P at time $t$ .

[6]

Find the total distance travelled by P when its velocity is increasing.

[4]

Question 2

HLPaper 1

The position vector of a particle, P, relative to a fixed origin O at time t is given by $\overrightarrow{OP} = \begin{pmatrix} \sin(t^2) \cos(t^2) \end{pmatrix}$

Find the velocity vector of P.

[2]

Show that the acceleration vector of P is never parallel to the position vector of P.

[5]

Question 3

HLPaper 2

A particle P moves along the x-axis. The velocity of P is v ms^(-1) at time t seconds, where v = -2t^2 + 16t - 24 for t ≥ 0.

Find the times when P is at instantaneous rest.

[2]

Find the magnitude of the particle's acceleration at 6 seconds.

[3]

Find the greatest speed of P in the interval 0 ≤ t ≤ 6.

[2]

The particle starts from the origin O. Find an expression for the displacement of P from O at time t seconds.

[4]

Find the total distance travelled by P in the interval 0 ≤ t ≤ 4.

[3]

Question 4

HLPaper 2

A car is being tested for its acceleration along a straight road. The acceleration of the car at time $t$ seconds is given by $a(t) = 2t - 4$ m/s².

Find the expression for the velocity of the car, given that the initial velocity at t = 0 is 5 m/s.

[3]

Determine the time when the car reaches its maximum velocity in the interval t ∈ [0, 10].

[5]

Find the total distance travelled by the car between t = 0 and t = 10.

[5]

Question 5

HLPaper 1

A particle moves along a straight line. Its displacement, $s$ metres, at time $t$ seconds is given by $s = t + \cos 2t,{\text{ }}t \geqslant 0$ . The first two times when the particle is at rest are denoted by ${t_1}$ and ${t_2}$ , where ${t_1} < {t_2}$ .

Find ${t_1}$ and ${t_2}$ .

[5]

Find the displacement of the particle when $t = {t_1}$

[2]

Question 6

HLPaper 2

At an archery tournament, a particular competition sees a ball launched into the air while anarcher attempts to hit it with an arrow.

The path of the ball is modelled by the equation

$(\begin{matrix} x \\ y \end{matrix}) = (\begin{matrix} 5 \\ 0 \end{matrix}) + t (\begin{matrix} u_{x} \\ u_{y} - 5 t \end{matrix})$

where $x$ is the horizontal displacement from the archer and $y$ is the vertical displacementfrom the ground, both measured in metres, and $t$ is the time, in seconds, since the ballwas launched.

$u_{x}$ is the horizontal component of the initial velocity
$u_{y}$ is the vertical component of the initial velocity.

In this question both the ball and the arrow are modelled as single points. The ball is launchedwith an initial velocity such that $u_{x} = 8$ and $u_{y} = 10$ .

An archer releases an arrow from the point $(0, 2)$ . The arrow is modelled as travelling in astraight line, in the same plane as the ball, with speed $60 {m s}^{- 1}$ and an angle of elevation of $10 °$ .

Find the initial speed of the ball.

[2]

Find the angle of elevation of the ball as it is launched.

[2]

Find the maximum height reached by the ball.

[3]

Assuming that the ground is horizontal and the ball is not hit by the arrow, find the $x$ coordinate of the point where the ball lands.

[3]

For the path of the ball, find an expression for $y$ in terms of $x$ .

[3]

Determine the two positions where the path of the arrow intersects the path of the ball.

[4]

Determine the time when the arrow should be released to hit the ball before the ballreaches its maximum height.

[4]

Question 7

HLPaper 2

A point Q moves in a straight line with velocity $v$ ms⁻¹ given by $v\left( t \right) = {{\text{e}}^{ - t}} - 8{t^2}{{\text{e}}^{ - 2t}}$ at timet seconds, wheret≥ 0.

Find the value of the acceleration of Q at time t₁.

Determine the first time t₁ at which Q has zero velocity.

Find an expression for the acceleration of Q at time t.

Question 8

HLPaper 2

A particle is tested in a laboratory setting, where its acceleration is given by $a(t) = 6t - 9$ m/s². The particle starts from rest at $t = 0$ , simulating conditions for studying motion dynamics.

Find the velocity function $v(t)$ .

[6]

Determine the time when the particle reaches a velocity of $18$ m/s.

[4]

Question 9

HLPaper 1

At $1 : 00 pm$ a ship is $1 km$ east and $4 km$ north of a harbour. A coordinate system is definedwith the harbour at the origin. The position vector of the ship at $1 : 00 pm$ is given by $(\begin{matrix} 1 \\ 4 \end{matrix})$ .

The ship has a constant velocity of $(\begin{matrix} 1.2 \\ - 0.6 \end{matrix})$ kilometres per hour ( ${km h}^{- 1}$ ).

Write down an expression for the position vector $r$ of the ship, $t$ hours after $1 : 00 pm$ .

[1]

Find the time at which the bearing of the ship from the harbour is $045 ˚$ .

[4]

Question 10

HLPaper 2

The position vector of a particle at time t is given by r = 3cos(3t)i + 4sin(3t)j. Displacement is measured in metres and time is measured in seconds.

Find an expression for the velocity of the particle at time t.

[1]

Hence find the speed when t = 3.

[4]

Find an expression for the acceleration of the particle at time t.

[1]

Hence show that the acceleration is always directed towards the origin.

[4]

For 0 ≤ t ≤ 10, find the time when the two particles are closest to each other.

[4]

Find the value of k.

[1]

At time k, show that the two particles are moving in the opposite direction.

[7]

At an archery tournament, a particular competition sees a ball launched into the air while anarcher attempts to hit it with an arrow.

The path of the ball is modelled by the equation

$(\begin{matrix} x \\ y \end{matrix}) = (\begin{matrix} 5 \\ 0 \end{matrix}) + t (\begin{matrix} u_{x} \\ u_{y} - 5 t \end{matrix})$

where $x$ is the horizontal displacement from the archer and $y$ is the vertical displacementfrom the ground, both measured in metres, and $t$ is the time, in seconds, since the ballwas launched.

$u_{x}$ is the horizontal component of the initial velocity
$u_{y}$ is the vertical component of the initial velocity.

In this question both the ball and the arrow are modelled as single points. The ball is launchedwith an initial velocity such that $u_{x} = 8$ and $u_{y} = 10$ .