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# Velocity

Bang! The rudder hits a rock and breaks. You can’t steer your boat! The propeller just moves you north now with $4\,\mathrm{m/s}$. The harbour is north-northeast, though… Luckily your boat has side rotors. You start the eastward rotor and adjust it, so that

• while the propeller continues to move you northward with $4\,\mathrm{m/s}$,
• the side rotor simultaneously moves you eastward with $3\,\mathrm{m/s}$.

Both speeds “take place” at the same time. Your motion is their combination, both a bit northward and a bit eastward simultaneously. This moves you north-northeast towards the harbour. Your motion is only one single motion north-northeast even though it is “made” from two speeds. Let’s as with position collect those two parts of the motion into a vector:

$$\vec v=\begin{pmatrix}4 \,\mathrm{m/s} \\3 \,\mathrm{m/s} \end{pmatrix}= \begin{pmatrix}4 \\3 \end{pmatrix} \mathrm{m/s}$$

• Previously, two positions in two different dimensions merged into one position with a direction and distance. This gave a vector form of ‘position’, and we added an arrow: $\vec s$.
• Now we similarly see two speeds/velocities merge into one velocity with a single direction and magnitude. They create a vector form of ‘velocity’. Let’s use the symbol: $\vec v$.

Exactly as with ‘position’, we now have two symbols for velocity: $v$ when direction is implied and not included, and $\vec v$ when it contains both direction and “size”/magnitude.

$$v=5\,\mathrm{m/s}\qquad \vec v=\left(\begin{matrix} 4 \\ 3 \end{matrix}\right) \,\mathrm{m/s}$$

Typically, people would call $v$ ‘speed’ and only use the term ‘velocity’ for $\vec v$. They might say:

• velocity is a vector; speed is its magnitude’, just as they would say:
• position is a vector; distance/length is its magnitude’.

‘Speed’ is then what the police officer cares about, while ‘velocity’ (‘speed with direction’) is relevant for a close-coming asteroid. This (rather arbitrary) distinction between ‘speed’ and ‘velocity’ in English is just tradition that became consensus.1 It is not universal and not all languages have two such words.2

Imagine a coordinate system placed south-to-north and west-to-east over the sea we are sailing at. The two speeds making up the velocity are northward and eastward, so along the $y$- and $x$-axes. Let’s label them $v_x$ and $v_y$ to tell them apart. While the velocity magnitude (speed) was ‘change in distance, $v=\mathrm ds/\mathrm dt$, the vector form similarly turns out to be: ‘change in position:

$\displaystyle \vec v=(v_x,v_y)=\biggl(\underbrace{\frac{\mathrm ds_x}{\mathrm dt}}_{v_x}, \underbrace{\frac{\mathrm ds_y}{\mathrm dt}}_{v_y} \biggr)=$$\displaystyle\frac{\overbrace{(\mathrm ds_x,\mathrm ds_y)}^{\mathrm d\vec s}}{\mathrm dt}=$$\displaystyle\underbrace{\frac{\mathrm d\vec s}{\mathrm dt}}_{\vec s’}=\vec s’$

So, velocity is the derivative of position to time, both in its vector form and in its non-vector form:

$v=s’\qquad$ and $\qquad \vec v=\vec s’$

Note that you can change your velocity without changing velocity magnitude (speed), such as when driving in circles.

References:

1. Vector analysis; a text-book for the use of students of mathematics and physics’ (book), J. Willard Gibbs & Bidwell Edwin Wilson, New York, C. Scribner’s Sons, 1901, babel.hathitrust.org/cgi/pt?id=mdp.39015000962285&view=1up&seq=149
2. Why are “speed” and “velocity” not given the same name?’ (web page, answer to forum post), Spareine, Physics Forums, 2012, www.physicsforums.com/threads/difference-between-speed-and-velocity.650579/#post-4164291 (accessed Sep. 23rd, 2019)
3. FysikABbogen 2’ (book, in Danish), Finn Elvekjær & Torben Benoni, Systime, 2013, fysikabbogen2.systime.dk/?id=c1649 (accessed Sep. 23rd, 2019), ISBN 9788761638182, chapter 5.2
4. Do “rapidez” and “velocidad” have similar technical meanings as “speed” and “velocity”?’ (web page, forum post), Nathaniel, Spanish Language Stack Exchange, 2015, spanish.stackexchange.com/questions/13158/do-rapidez-and-velocidad-have-similar-technical-meanings-as-speed-and-vel (accessed Sep. 26th, 2019)
5. Rapidez y velocidad’ (web page, in Spanish), educaplus.org, www.educaplus.org/movi/2_5velocidad.html (accessed Sep. 23rd, 2019)
6. Speed and Velocity in German’ (web page, forum discussion), Jannik Pitt, German Language Stack Exchange, 2016, german.stackexchange.com/questions/32366/speed-and-velocity-in-german (accessed Sep. 23rd, 2019)
7. Speed and Velocity in Russian’ (web page, forum post), Steeven Hegelund Spangsdorf, Russian Language Stack Exchange, 2019, russian.stackexchange.com/questions/20572/speed-and-velocity-in-russian/20574 (accessed Sep. 23rd, 2019)