Motion in two dimensions

In a nutshell

When a vector quantity, such as a force, is given in component form, you can solve problems in $2$ dimensions. Force, velocity and acceleration are all vectors, and you should be able to solve problems with vectors given in component form. You can calculate the magnitude and direction of a vector.

Equations

Description	equation
Newton's second law	$F=ma$
Magnitude of a vector $x{\bf i} + y {\bf j}$	$\sqrt{x^2+y^2}$
Direction of a vector $x{\bf i} + y {\bf j}$	$\tan^{-1}\dfrac{y}x$

Variable Definitions

Quantity Name	Symbol	Unit name	unit
$Force$	$F$	$Newton$	$N$
$Mass$	$m$	$Kilogram$	$kg$
$Acceleration$	$a$	$Metres\space per\space second\space squared$	$ms^{-2}$
$Velocity$	$v$	$Metres \ per \ second$	$ms^{-1}$

Force and acceleration as vectors

As seen before, force vectors can be written in component from $\bold {i,j}$ , where $\bold i$ and $\bold j$ are perpendicular to each other. Velocity and acceleration are also vectors, so they can also be written in terms of $\bold i$ and $\bold j$ .

Below is a diagram of an object being acted on by acceleration vector $\bold a$ .

Maths; Forces and motion; KS5 Year 12; Motion in two dimensions

The vector $\bold a$ is separated into components - a horizontal component of value $x$ and a vertical one of value $y$ . The vectors direction makes an angle $\theta$ with the $x$ -axis. You can write this information as such:

$\begin{aligned}{\bf a}&=(x{\bf i} + y {\bf j})\\\\\tan\theta&=\dfrac{y}{x}\\\end{aligned}$

Note- $(x{\bf i} + y {\bf j})$ can also be written in the form $\begin{pmatrix} x \\ y \end{pmatrix}$

Using Pythagoras' theorum and trigonometry, you can work out the magnitude of $\bold a$ and direction dictated by the angle $\theta$ .

Magnitude	$\|\textbf{a}\|=\sqrt{x^2+y^2}$
Direction	$\theta=\tan^{-1}\dfrac{y}{x}$

Example 1

A resultant force $(2{\bf i} + 6 {\bf j})N$ acts upon an object of mass $0.25$ $\space kg$ . What is the magnitude and direction of the object's acceleration $\bold a$ ? Give your answer correct to one d.p

Find the value for $\bold a$ in component form.

$\begin{aligned}F&=ma\\F&=(2{\bf i} + 6 {\bf j})N\\m&=0.25\space kg\\(2{\bf i} + 6 {\bf j})&=0.25a\\a&=(8{\bf i} + 24 {\bf j})\end{aligned}$

Find the magnitude and direction of $\bf a$ .

$\begin{aligned}|a|&=\sqrt{8^2+24^2}\\|a|&=\sqrt{64+576}\\|a|&=\sqrt{640}\\|a|&=8\sqrt10\\\tan\theta&=\dfrac{24}{8}\\\tan\theta&=3\\\theta&=\tan^{-1}3\\\theta&=71.6\degree\end{aligned}$

$Magnitude = \underline{8\sqrt{10}}\space ms^{-2}$ $Direction=\underline{71.6} \degree$