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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…

The radian unit

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…

Vector addition

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…

Deeper derivatives

When speed $v=s’$ itself is changing, $v’=s^{\prime\prime}$, we reach a deeper derivative. In Leibniz notation: $\displaystyle v’=\frac{\mathrm dv}{ \mathrm dt}=\frac{ \mathrm d\overbrace{( \mathrm ds/ \mathrm…

Derivatives

On a trip, the car speedometer shows the speed right now. Instantly. Instantaneously. If your car could measure the time for every 100 metres, it…

Fundamental units

There do exist other position units than the metre, for example miles, inches, yards and feet, other time units than the second, for example minutes,…

Subtraction

The bagel store is 5 kilometres from home. School is 3 km from home. Drive on your bike from school to the bagel store, and…

Division

The World’s fastest on a computer writes a bit more than 210 words per minute; more than 210 words during those 60 seconds that one…

Curvature

A hilltop dome peaks in a highest point. A valley on the other hand “peaks” in a lowest point.

Signs & negative numbers

A good part-time salary of 2000 € per month keeps your bank balance above zero. That makes you happy; that makes you positive about the…

Geometry

2D shapes of course have area, and 3D shapes volume. How do we find these sizes of shapes? It turns out that they can be…

Natural constants & pi

What is the size of your dinner pizza? It’s a bit hard to measure a round area; can’t we instead measure something else on the…

Powers & exponents

In a game of ‘Double or Quits’ you just won 4 times in a row. The $200\,€$ you started out with are doubled 4 times: $$200\,€\cdot…

Multiplication & algebra

You have to take a detour to work this morning and ‘that trip is 3 times longer’ (3 times the usual route): $$s_\mathrm{detour}=3\;s_\mathrm{usual}$$

Coordinate systems

A crucial detail for pointing out a position such as $\vec s=(15,20)\, \mathrm{cm}$ on a map is where to start. And also, which ways to…

Vectors

Let’s give a name to properties that are collections of numbers: let’s call them vectors. Position is hereby a vector. We can call it the…

Prefixes

There are about $400\,000\,\mathrm m$ by car from Copenhagen to Aalborg and $2\,600\,000\,\mathrm s$ in a month, while the thickness of a pencil stroke is…

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…