A-level · Kinematics

Motion in two dimensions with vectorsExam-style questions

Why the a → v → r calculus chain works component-by-component in 2D, that speed is the magnitude of the velocity vector, and how "parallel to" and "perpendicular to" a direction become simple component conditions.

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33 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
[In this question $\mathbf{i}$ and $\mathbf{j}$ are horizontal unit vectors directed due east and due north respectively and position vectors are given relative to a fixed origin $O$ .]

Two canoes, $A$ and $B$ , are moving across a lake. Each canoe moves in a straight line at a constant velocity, and the canoes are modelled as particles.

Canoe $A$ passes the point with position vector $(-4\mathbf{i} + 21\mathbf{j})$ m at time $t = 5$ seconds, and it passes the point with position vector $(26\mathbf{i} + 6\mathbf{j})$ m at time $t = 20$ seconds.
(a) Find the velocity of $A$ , giving your answer in terms of $\mathbf{i}$ and $\mathbf{j}$ .

(2 marks)
(b) Find, in terms of $t$ , $\mathbf{i}$ and $\mathbf{j}$ , the position vector of $A$ at time $t$ seconds.

(2 marks)

Canoe $B$ moves with constant velocity $(-2\mathbf{i} + \mathbf{j})\ \text{m s}^{-1}$ . When $t = 0$ , $B$ is at the point with position vector $(40\mathbf{i} - 10\mathbf{j})$ m.
(c) Find the bearing of $A$ from $B$ when $t = 25$ , giving your answer to the nearest degree.

(3 marks)
Level 3 · 7 marksTry it
2
A particle moves in a plane so that at time $t$ seconds its position vector is $\mathbf{r} = (t^3 - 2t)\,\mathbf{i} + (3t^2 - t)\,\mathbf{j}$ metres. Find
(a) its velocity at time $t$ ,
(b) its acceleration at time $t$ , and (c) its speed when $t = 1$ .
Level 3 · 5 marksTry it
3
A particle moves with constant acceleration $\mathbf{a} = (-\mathbf{i} + 2\mathbf{j})\ \text{m s}^{-2}$ . At $t = 0$ its velocity is $\mathbf{u} = (4\mathbf{i} - 3\mathbf{j})\ \text{m s}^{-1}$ . Find
(a) its velocity when $t = 3$ ,
(b) its speed when $t = 3$ , and (c) the value of $t$ at which it is moving parallel to $\mathbf{j}$ .
Level 3 · 4 marksTry it
4
A particle moves in a plane so that at time $t$ seconds its position vector is

$\mathbf{r} = (3t^{2} - 1)\,\mathbf{i} + (2t + 4)\,\mathbf{j} \text{ metres}$

Find the speed of the particle when $t = 2$ , giving your answer in the form $k\sqrt{37}$ m s $^{-1}$ . (4 marks)
Level 3 · 4 marksTry it
5
A particle has initial velocity $(2\mathbf{i} - 3\mathbf{j})$ m s $^{-1}$ and moves with constant acceleration $(4\mathbf{i} + \mathbf{j})$ m s $^{-2}$ .

Find its velocity after $3$ seconds, and describe the direction of motion at that instant. (3 marks)
Level 3 · 3 marksTry it

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