## 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…

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…

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…

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…

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…

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…

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…

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

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…

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…

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…

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…

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…

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,…

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…

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…

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…

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

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

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,…

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…

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…

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…

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…

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…

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,…

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…

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…

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

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…

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…

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…

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…

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

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…

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…

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…

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

‘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…

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…