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 you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

[In this question, $\mathbf{i}$ and $\mathbf{j}$ are horizontal perpendicular unit vectors.]

A particle $P$ moves on a smooth horizontal plane. At time $t$ seconds, $t \geqslant 0$ , the position vector of $P$ relative to a fixed origin $O$ is $\mathbf{r}$ metres, where

$\mathbf{r} = (2t^2 - 5t)\mathbf{i} + (t^3 - 6t^2 + 9t)\mathbf{j}$
(a) Find the speed of $P$ at the instant when $P$ is moving in the direction of the vector $\mathbf{i}$

(5 marks)
(b) Find the exact magnitude of the acceleration of $P$ when $t = 3.5$

(4 marks)
(c) Show that $P$ never returns to $O$

(2 marks)
Level 4 · 11 marksTry it
2
[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 $O$ .]

An oil platform at sea is modelled as a fixed point $P$ with position vector $(13\mathbf{i} + 7\mathbf{j})$ km. Ships are not allowed to enter a circular safety zone of radius $2$ km centred on $P$ .

A cargo ship $C$ is moving with constant velocity $(-12\mathbf{i} + 16\mathbf{j})\ \text{km h}^{-1}$ . At 05:30 the ship is at the point with position vector $(31\mathbf{i} - 14\mathbf{j})$ km. The ship is modelled as a particle.
(a) Find the speed of $C$ .

(2 marks)
(b) Find, in terms of $\mathbf{i}$ , $\mathbf{j}$ and $t$ , the position vector of $C$ at time $t$ hours after 05:30

(2 marks)
(c) (i) Find the value of $t$ for which the distance between $C$ and $P$ is least.

(ii) Hence determine whether $C$ will enter the safety zone if it continues on its present course.

(7 marks)
Level 4 · 11 marksTry it
3
[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 helicopters, $A$ and $B$ , are flying at the same constant height. Each helicopter flies in a straight line at a constant velocity, and the helicopters are modelled as particles moving in the same horizontal plane.

Helicopter $A$ flies directly over a beacon with position vector $(-20\mathbf{i} + 41\mathbf{j})$ km at 09:40 and directly over a second beacon with position vector $(52\mathbf{i} + 11\mathbf{j})$ km at 10:00

The position vector of $A$ , $t$ hours after 09:40, is $\mathbf{r}_A$ km.
(a) Show that $\mathbf{r}_A = (-20 + 216t)\mathbf{i} + (41 - 90t)\mathbf{j}$

(4 marks)
(b) Find the speed of $A$ in $\text{m s}^{-1}$

(2 marks)

Helicopter $B$ flies with constant velocity $(-120\mathbf{i} - 50\mathbf{j})\ \text{km h}^{-1}$ . At 09:40, $B$ is at the point with position vector $(136\mathbf{i} + 16\mathbf{j})$ km.

The paths of the two helicopters cross at the point $X$ .
(c) Find the position vector of $X$ .

(5 marks)
Level 4 · 11 marksTry it
4
A particle $P$ moves in a horizontal plane. The velocity of $P$ at time $t$ seconds is $\mathbf{v}\ \text{m s}^{-1}$ , where $\mathbf{v} = (2t^2 - t - 7)\,\mathbf{i} + (t^2 - 5t - 2)\,\mathbf{j}, \qquad t \ge 0.$
(a) Find the initial speed of $P$ .

(3 marks)
(b) Find the value of $t$ for which the velocity of $P$ is parallel to the vector $(\mathbf{i} - \mathbf{j})$ .

(2 marks)
(c) Find, in terms of $t$ , the acceleration of $P$ .

(2 marks)
(d) Find the time at which $P$ is accelerating in a direction perpendicular to $\mathbf{j}$ .

(2 marks)
Level 4 · 9 marksTry it
5
A particle moves in a plane with velocity $\mathbf{v} = (6t - 4)\,\mathbf{i} + 3t^2\,\mathbf{j}\ \text{m s}^{-1}$ . When $t = 0$ its position vector relative to a fixed origin $O$ is $(2\mathbf{i} + \mathbf{j})$ m. Find
(a) its position vector at time $t$ ,
(b) its position vector when $t = 2$ , and (c) its distance from $O$ when $t = 2$ .
Level 4 · 5 marksTry it

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