SL 5.9—Kinematics problems Questionbank

Practice SL 5.9—Kinematics problems with authentic IB Mathematics Analysis and Approaches (AA) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like functions and equations, calculus, complex numbers, sequences and series, and probability and statistics. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

SL & HLPaper 1

A car accelerates from rest with an acceleration given by $a(t) = 4t - 2$ , where $a$ is in meters per second squared and $t$ is in seconds.

Find the velocity of the car as a function of time.

[4]

Calculate the distance traveled by the car in the first 5 seconds.

[5]

Question 2

SL & HLPaper 1

The velocity of a particle is given by $v(t) = 4t - 5$ .

Find the displacement of the particle from $t = 0$ to $t = 4$ .

[5]

Question 3

SL & HLPaper 1

A particle is moving along the curve defined by the function $y = x^3 - 3x + 2$ . The position of the particle at any time $t$ is given by $(x(t), y(t))$ where $x(t)$ and $y(t)$ are the coordinates of the particle at time $t$ . The following questions explore the motion of the particle along the curve.

Find the velocity of the particle at time $t$ in terms of $x(t)$ and $\frac{dx}{dt}$ .

[2]

Determine the acceleration of the particle at time $t$ in terms of $x(t)$ , $\frac{dx}{dt}$ , and $\frac{d^2x}{dt^2}$ .

[3]

At what points on the curve is the velocity of the particle zero?

[2]

Question 4

HLPaper 1

An object travels along a linear path. Its displacement, $s$ metres, at time $t$ seconds is given by $s = t + \cos 2t$ , $t \geqslant 0$ . The first two times when the object is stationary are denoted by $t_1$ and $t_2$ , where $t_1 < t_2$ .

Find $t_1$ and $t_2$ .

[5]

The displacement of the object when $t = {t}_1$ .

[2]

Question 5

HLPaper 1

In this question, all lengths are in metres and time is in seconds. Consider two particles, $A_1$ and $A_2$ , which start to move at the same time. Particle $A_1$ moves in a straight line such that its displacement from a fixed-point is given by $s(t) = 10 - \frac{7}{4}t^2$ , for $t \geq 0$ .

Find an expression for the velocity of ${A}_1$ at time ${t}$ .

[2]

Particle $A_2$ also moves in a straight line. The position of $A_2$ is given by $r=-\frac{1}{6}+t\left(\begin{array}{c}4\\-3\end{array}\right).$ The speed of $A_1$ is greater than the speed of $A_2$ when $t>q$ . Find the value of $q$ .

[5]

Question 6

SLPaper 2

In this question, all lengths are in metres and time is in seconds. A particle moves in a straight line such that its displacement from a fixed point is given by $s(t)=5 t-\frac{1}{3} t^{3}$ , for $t \geq 0$ .

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

[2]

Find the time $t$ when the particle is at rest.

[2]

Calculate the acceleration of the particle at $t=2$ .

[2]

Question 7

SLPaper 2

In this question, all lengths are in metres and time is in seconds. A particle moves along a straight line with acceleration given by $a(t)=6 \cos \left(\frac{\pi}{4} t\right)-\frac{2}{(t+1)^{2}}$ , for $t \geq 0$ . The particle starts from rest at the origin at $t=0$ . At times $t_{1}, t_{2}, t_{3}$ , where $0<t_{1}<t_{2}<t_{3}<4$ , the particle's velocity is zero, and the corresponding displacements $s_{1}, s_{2}, s_{3}$ form an arithmetic sequence.

Find an expression for the velocity $v(t)$ of the particle.

[4]

Determine the times $t_{1}, t_{2}, t_{3}$ when the particle is at rest in $0 \leq t<4$ .

[4]

Show that $s_{2}-s_{1}=s_{3}-s_{2}$ , and find the common difference of the arithmetic sequence.

[5]

Calculate the total distance travelled by the particle from $t=0$ to $t=4$ .

[5]

Question 8

SLPaper 2

In this question, all lengths are in metres and time is in seconds.

Two particles, A and B , move along the same straight line. Particle A has velocity $v_{A}(t)=t e^{-0.5 t}+2 \cos \left(\frac{\pi}{6} t\right)$ , for $0 \leq t \leq 12$ . Particle B has displacement $s_{B}(t)=6 t-\frac{1}{3} t^{2}$ , for $t \geq 0$ . Both particles start at the origin at $t=0$ .

Find the time when particle A's acceleration is zero in $0 \leq t \leq 12$ .

[4]

Determine the times when particles A and B have the same velocity.

[4]

Find the time when the distance between particles A and B is minimized in $0 \leq$ $t \leq 12$ .

[5]

Question 9

SLPaper 1

A particle moves along a straight line with velocity $v(t)=2 t^{2}-8 t+6$ , for $0 \leq t \leq 5$ , where velocity is in metres per second and time is in seconds. The particle is at the origin at $t=0$ .

Find the time when the particle is at rest.

[3]

Calculate the displacement of the particle at $t=5$ .

[3]

Sketch the velocity-time graph, indicating the intercepts and vertex.

[3]

Question 10

SLPaper 2

A particle moves along a straight line with velocity $v(t)=t^{2} e^{-t}$ , where $t \geq 0$ is time in seconds and $v$ is in $\mathrm{m} / \mathrm{s}$ .

Show that the acceleration of the particle is given by $a(t)=t e^{-t}(2-t)$ , and find the times when the acceleration is zero.

[4]

Prove that the position-time graph has exactly one point of inflection, and find its coordinates.

[5]

Find the maximum velocity of the particle, and determine whether the positiontime graph is concave up or concave down at this point. marks]

[4]