 ## Cross-product

Right now we have two versions of the torque formula, $\tau=r_\perp\;F$ and $\tau=r\;F_\perp$, that include $\perp$-symbol, possibly sines etc. All we wanted was to ‘multiply…

## Pseudovectors

To get to the other side of the block, should you go ‘right and then left’ or ‘left and then right’? It of course doesn’t…

## Rotations as vectors

Linear properties point backwards or forwards whereas angular properties “point” clockwise or counterclockwise. While the backwards/forwards feeling is along something – a direction that defines…

## Negative unit exponents

Position $s$ is in units of ‘metres’ $\mathrm m$, speed $v$ is in ‘metres-per-second’ $\mathrm{m/s}$, and acceleration $a$ is in ‘metres-per-second-squared’ $\mathrm{m/s^2}$. Now, what are…

## Sine & cosine values

It is not easy to figure out the actual values of cosine and sine to different angles. Only a few angles are obvious: an angle…

To measure how long the Roman Colosseum is around (circumference), the archaeologist can with a measuring wheel choose to walk all the way around. But…

## Inequations

Can you guess how many cinnamon buns I ate at work today? Let’s symbolise ‘the number of cinnamon buns’ with an $x$ for now. You…

## Proportionality

At the bakery, you order a portion of layered birthday cake for 3 €. It is pretty good. You order 3 more. You of course…

## Trigonometry

The vector components can of course give the original vector if we just add them up: Draw $\vec F_y$, continue with $\vec F_y$ and you…

## Vector components

With the sine and cosine “standard” invented, we can now more easily say that a jumbo jet requiring $F=10\,000\,\mathrm N$ of force at a certain…

## Sine & cosine

An airplane taking off needs a horizontal force about 5 times larger than its vertical lift force. We can write that as: $F_x=5F_y$. This is…

## Zero vector

The towed boat doesn’t accelerate perpendicular to the stream it is being pulled along. In other words: $a_x=0$. Furthermore, maybe the forwards-pull just balances out…

## The Newton unit

To talk about ‘how much force’ there is, we of course need a unit for force. Should that be a fundamental unit? Nope. Force is…

## Sigma summation

We could easily have many more forces to add. While towing your boat, the stream might get stronger and a stormy wind may appear, so…

To pull your boat up the stream, you attach a rope and pull from the shore. This will pull the boat towards the shore, though,…

## Force diagrams

When drawing a force diagram like the following, what are we then not drawing? What is left out? All the forces that influence the book…

## Square root

6 apples in one bag. We need many, so we’ll take six bags each: $$6\cdot 6=6^2=36$$ At home, when your family says ‘that’s a lot…

## Pythagoras’ theorem

How far away is the orange after, say 2 seconds of flight? We are looking for the distance $s$ here; not the coordinates $s_x$ or…

## Linear & parabolic

Certain shapes on graphs are easy to recognise. Let’s imagine a thrown orange.

## Graphs

An equation for a motion is, shall we say, not exactly visually pleasing. Just a ‘mathematical description’ or ‘recipe’ – or model – of a…

## Rules of algebra

Let’s rearrange some more: \begin{align}s_y&=v_{y,0}\left(\frac{s_x}{v_{x,0}}\right)-\frac12 g \left(\frac{s_x}{v_{x,0}}\right)^2\\ &=\boldsymbol{\frac{ v_{y,0} s_x}{v_{x,0}}}-\frac12 g \left(\frac{s_x}{v_{x,0}}\right)^2 \\ &= \frac{ v_{y,0} s_x}{v_{x,0}} -\frac12 g \boldsymbol{\frac{s_x^2}{v_{x,0}^2}}\end{align} Here, in the first step,…

## Equation manipulation

One thing is knowing how long time it takes for the orange to fly sideways. We reached the expression $t=\frac{s_x}{v_{x,0}}$ in the previous skill as…

## Equation rearrangement

Previously (in the Throwing & free falling skill) we arrived at these two equations for our flying orange-cannon shot orange: $\displaystyle s_y=v_{y,0}t- \frac12 gt^2\qquad$ and…

## Addition & the four basic operations

You have 3 apples. Your friend gives you 6 apples more. How many apples do you have in total? 9, of course. ‘3 and 6…

## Decimals

You are maybe just below 2 metres tall. That is, 1 metre and some – but not 2. To be more precise you could choose…

## Units

‘The chair is 2 away’ and ‘the popcorn needs 10 more’ make no sense to say. But ‘the chair is 2 metres away’ and ‘the…

## Dimensions

A train is restricted to its track and can only move back and forth (along one direction). It moves along a length. A car has…