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’*:

$\displaystyle \vec v=(v_x,v_y)=\biggl(\underbrace{\frac{\mathrm ds_x}{\mathrm dt}}_{v_x}, \underbrace{\frac{\mathrm ds_y}{\mathrm dt}}_{v_y} \biggr)=$^{3}$\displaystyle\frac{\overbrace{(\mathrm ds_x,\mathrm ds_y)}^{\mathrm d\vec s}}{\mathrm dt}=$^{4}$\displaystyle\underbrace{\frac{\mathrm d\vec s}{\mathrm dt}}_{\vec s’}=\vec s’$

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:

- ‘
**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 - ‘
**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) - ‘
**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 - ‘
**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) - ‘
**Rapidez y velocidad**’ (web page, in Spanish), educaplus.org, www.educaplus.org/movi/2_5velocidad.html (accessed Sep. 23rd, 2019) - ‘
**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) - ‘
**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)