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

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Variable Definitions

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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 $$\bold a$$.

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 $$\theta$$.

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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 $$\bf a$$.

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

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

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