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SL 5.9—Kinematics problems - IB Questionbank | RevisionDojo

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 displacement of the car in the first 5 seconds.

[5]

Question 2

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]

Find the coordinates of the points on the curve where the vertical component of the velocity is zero, assuming that $\frac{dx}{dt} \neq 0$ .

[2]

Question 3

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]

Find the displacement of the object when $t = t_1$ .

[2]

Question 4

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 $\mathbf{r} = t\begin{pmatrix} 4 \\ -3 \end{pmatrix}$ . The speed of $A_1$ is greater than the speed of $A_2$ when $t > q$ . Find the value of $q$ .

[5]

Question 5

SLPaper 2

An object moves in a straight path such that its speed, $v \text{ m s}^{-1}$ , at time $t$ seconds is given by

$v=4t^2-6t+9-2\sin(4t), \quad 0 \leq t \leq 1$

The object’s acceleration is zero at $t=T$ .

Determine the value of $T$ .

[2]

Let $d_1$ be the distance covered by the object from $t=0$ to $t=T$ and let $d_2$ be the distance covered by the object from $t=T$ to $t=1$ . Demonstrate that $d_2 > d_1$ .

[3]

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

SL & HLPaper 1

The velocity of a particle is given by $v(t) = 4t - 5$ , where $t$ is in seconds and $v$ is in $\text{m s}^{-1}$ .

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

[5]

Question 8

HLPaper 2

A small marble is free to move along a smooth wire in the shape of the curve $y=\frac{10}{3-2e^{-0.5x}}, \quad x \geq 0$

At the point on the curve where $x=4$ , it is given that $\frac{dy}{dt}=-0.1\,\mathrm{m\,s^{-1}}$ .

Find the value of $\frac{dx}{dt}$ at this instant.

[3]

Find an expression for $\frac{dy}{dx}$ .

[3]

Question 9

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 $a(t)=6 \cos(3t)-\frac{2}{(t+1)^{2}}$ , 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.

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 $0 < t < 4$ .

[4]

Calculate the corresponding displacements $s_{1}, s_{2}, s_{3}$ of the particle at times $t_{1}, t_{2}, t_{3}$ .

[5]

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

[5]

Question 10

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

[4]

Find the time when particles A and B have the same velocity.

[4]

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

[5]

SL 5.9—Kinematics problems Questionbank