Projectiles

Example: Projectiles in Vector Notation

We can also use vectors to make projectile motion much neater.

So, for example, say a ball is thrown off of a cliff with a velocity of $(15textbf{i} + 7textbf{j})text{ ms}^{-1}$ with $textbf{i}$ its horizontal velocity, and $textbf{j}$ its upward vertical velocity. Assume that the ball accelerates due to gravity and experiences no air resistance. Given it is in the air for $textcolor{purple}{t} = textcolor{purple}{5}text{ seconds}$, how tall is the cliff, what horizontal distance does the ball travel and what is its final velocity?

Assume $g = 10text{ ms}^{-2}$.

[4 marks]

$textcolor{limegreen}{underline{s}} = underline{u}textcolor{purple}{t} + dfrac{1}{2}textcolor{blue}{underline{a}}textcolor{purple}{t}^2$

gives

$textcolor{limegreen}{underline{s}} = textcolor{purple}{5}(15textbf{i} + 7textbf{j}) + dfrac{textcolor{purple}{25}}{2}(textcolor{blue}{-10textbf{j}}) = textcolor{limegreen}{75textbf{i} – 90textbf{j}}$

So, the ball travels $textcolor{limegreen}{75}text{ m}$ horizontally, and the cliff is $textcolor{limegreen}{90}text{ m}$ tall.

$textcolor{red}{underline{v}} = underline{u} + textcolor{blue}{underline{a}}textcolor{purple}{t}$

gives

$textcolor{red}{underline{v}} = (15textbf{i} + 7textbf{j}) – (textcolor{blue}{10} times textcolor{purple}{5})textbf{j} = textcolor{red}{15textbf{i} – 43textbf{j}}text{ ms}^{-1}$