You are driving a steady $20\,\mathrm{m/s}$ ($72 \,\mathrm{km/hr}$) but a car behind you is rushing. You push the gas pedal and slowly speed up by gaining 2 more metres-per-second every second.

Those additional 2 metres-per-second extra per second – we could write them as $2 \,\mathrm{m/s\,/s}$ or simplify it to $ 2\,\mathrm{m/s^2} $ – are a *change in the speed*. Let’s name ‘change in speed’: **acceleration**.

- Speed is ‘how fast’, the change in distance, and
- acceleration is ‘how
**much faster**’, the change in speed.

As we could move sideways and forwards simultaneously – two speeds $v_x$ and $v_y$ that combine into one velocity vector $\vec v$ – likewise accelerations in several directions, $a_x$ and $a_y$, combine into one **acceleration vector** $\vec a$.^{1}

$$\vec s= (s_x,s_y)\qquad

\vec v=(v_x,v_y) \qquad

\vec a=(a_x,a_y)$$

Since acceleration is ‘a change in velocity’, which is ‘a change in position’, acceleration must be ‘a change *in the change* in position’; a *double change*:

$$a=v’=s^{\prime \prime } \qquad \vec a=\vec v’=\vec s^{\prime \prime }$$

‘Speeding up’ or ‘slowing down’^{2} as well as ‘turning’ are changes in speed or velocity, and thus require an acceleration.

Note an odd fact: Our bodies do not feel position. And also not velocity. **We only feel acceleration**. You can sit in a bus and drink your coffee undisturbed, unaware of whether the bus is standing still or is at full speed down the motorway. But the moment it turns, brakes or speeds up, then your coffee flies all over the place, and you feel squeezed into the seat or flung around.^{[2]}

References:

- ‘
**Sears and Zemansky’s Univesity Physics with Modern Physics**’ (book),*Hugh D. Young & Roger A. Freedman*, Pearson Education, 13th ed., 2012 - ‘
**Beyond velocity and acceleration: jerk, snap and higher derivatives**’ (article),*David Eager, Ann-marie Pendrill and others*, European Journal of Physics, IOP Publishing, vol. 37, issue 6, 2016, www.iopscience.iop.org/article/10.1088/0143-0807/37/6/065008, DOI 10.1088/0143-0807/37/6/065008