Year 12 Maths Extension 1 (2027) Vectors

Motion in vector form 2D

Q: How do you find position from acceleration?

Integrate acceleration to velocity to position, fixing constants from the initial conditions.

Q: How do you find the Cartesian path?

Eliminate t between x of t and y of t.

Q: What is speed?

The magnitude of the velocity vector.

20 practice questions 2 video lessons Theory + worked examples

Study motion in vector form for NSW Year 12 Mathematics Extension 1. Writing position as a vector function of time describes a moving object in the plane, with its path traced out by parametric equations.

You will learn to differentiate and integrate to find velocity and acceleration, convert between vector and Cartesian paths, and solve relative-velocity problems such as navigating a crosswind or cross-current — practical modelling in the Extension 1 course.

Practice 20 questions

Practice questions

Every question with a fully worked solution.

Start practising

Watch 2 video(s)

Find Initial Position, Velocity Vector, and Speed From Position Vector Equation (2D) Watch
Calculus 16.3 Motion Along 2D Curves Watch

Theory

Motion in a plane uses $r (t) = x (t) i + y (t) j$ , with $v = \dot{r}$ and $a = \dot{v}$ ; integrate to reverse. This NSW Year 12 Mathematics Extension 1 topic (NESA outcome ME1-12-02) covers Cartesian paths and relative velocity.

Motion in a plane is described by a position vector $r (t) = x (t) i + y (t) j$ . Differentiating gives velocity and acceleration; integrating reverses the process.

v = \dot{r} = \dot{x} i + \dot{y} j, a = \dot{v} = \ddot{x} i + \ddot{y} j .

Speed is $| v |$ ; the distance between two positions is $| Δ r |$ . Eliminating $t$ from $x (t)$ , $y (t)$ gives the Cartesian path. For a crosswind or current, the resultant velocity is the vector sum.

NESA link. Part of the Year 12 Introduction to vectors focus area, outcome ME1-12-02 ("operates with 2D and 3D vectors and uses 2D vectors to solve problems involving motion in two dimensions") with MAO-WM-01.

Resultant velocity is the vector sum of boat and current.

Velocity

v = \dot{r}

is tangent to the path.

speed = | v |, r = r_{0} + t v (constant velocity) .

speed = |v|; constant velocity r = r0 + t v

Reverse with integration, adding a constant vector fixed by initial conditions:

a \overset{\int}{\to} v \overset{\int}{\to} r .

integrate a to get v, integrate v to get r

Bearings. Measure from north (the $j$ direction), e.g. N $θ$ E, and give speed as the magnitude of the resultant.

How to solve motion problems

Differentiate $r (t)$ for velocity, again for acceleration.
Integrate $a \to v \to r$ , fixing constants from initial conditions.
Cartesian path: eliminate $t$ between $x (t)$ and $y (t)$ .
Resultant velocity: add the vectors; magnitude is speed, direction is the bearing.

Example 1 — Velocity and acceleration

A particle has

r (t) = 2 t^{2} i + (t^{2} - 4 t) j

. Find

v (t)

and

a (t)

Solution

$v$	$=$	$4 t i + (2 t - 4) j$
$a$	$=$	$4 i + 2 j$

v = 4t i + (2t - 4) j, a = 4i + 2j

$v = 4 t i + (2 t - 4) j$ , $a = 4 i + 2 j$ .

Example 2 — Cartesian path

A particle has

r (t) = (2 + t) i + t^{2} j

. Find the Cartesian equation of its path.

Solution

$x = 2 + t$ , so $t = x - 2$ .

y

=

t^{2} = (x - 2)^{2}

path: y = (x - 2)^2

The path is $y = (x - 2)^{2}$ .

Example 3 — Integrate to position

a (t) = 6 t i + 2 j

, with

v (0) = i + 3 j

and

r (0) = 0

. Find

v (t)

and

r (t)

Solution

$v$	$=$	$(3 t^{2} + 1) i + (2 t + 3) j$
$r$	$=$	$(t^{3} + t) i + (t^{2} + 3 t) j$

r = (t^3 + t) i + (t^2 + 3t) j

$r = (t^{3} + t) i + (t^{2} + 3 t) j$ .

Example 4 — Crosscurrent

A boat heads north at

12

m/s while a current flows east at

5

m/s. Find the resultant speed and bearing.

Solution

Resultant $v = 5 i + 12 j$ .

$\| v \|$	$=$	$\sqrt{5^{2} + 12^{2}} = 13$
$θ$	$=$	$\tan^{- 1} \frac{5}{12} \approx 23^{\circ}$

speed = 13 m/s, bearing approx N23E

Speed $13$ m/s, bearing $\approx$ N $23^{\circ}$ E.

Common pitfalls

Speed vs velocity. Speed is the magnitude

| v |

, a scalar; velocity is a vector.

Constant of integration. Integrating gives a constant vector, fixed by the initial conditions.

Eliminating $t$ . Make

t

the subject of the simpler component, then substitute.

Bearings. Measure from north, not from the

x

-axis.

Frequently asked questions

How do you find velocity and acceleration?

Differentiate the position vector once for velocity, twice for acceleration.

How do you find position from acceleration?

Integrate $a \to v \to r$ , fixing constants from the initial conditions.

How do you find the Cartesian path?

Eliminate $t$ between $x (t)$ and $y (t)$ .

What is the resultant velocity?

The vector sum of the object's velocity and the wind or current.

What is speed?

The magnitude of the velocity vector, $| v |$ .

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Motion in vector form 2D

📖 Theory

How to solve motion problems

Common pitfalls

Frequently asked questions

Theory