Could velocity be taken as fundamental instead of time? In physics time and length are taken as fundamental in the SI system and, as it seems, in the thinking of physicists. Could one instead take velocity, with c as its unit, together with length as fundamental and then understand time by dimensional analysis in terms of l/v (length divided by velocity)? If not, why not?
 A: You can work with any system of units you want, but some units are more convenient than others. In everyday physics we regularly have to calculate dependance on time, i.e. $df/dt$, and it's a great deal easier to do this if we express our function $f$ as a function of time than as a function of velocity. It's pretty rare we want to calculate $df/dv$.
So taking length and velocity as the fundamental units would just make our lives harder.
Having said this, and maybe this is what you meant, the SI unit the metre is currently defined as the distance travelled by light in a vacuum in 1/299,792,458 seconds. So we are using the velocity $c$ to define the metre. The second remains defined as a time i.e. as the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom.
And just to add to the mix, in relativity we conventionally convert time to a distance by multiplying by the velocity $v$, though to be awkward we define $c$ as $1$.
A: You need three quantities to fully define all mechanical units, like $({\rm Mass},\,{\rm Length},\,{\rm Time})$ or $({\rm Force},\,{\rm Length},\,{\rm Time})$.
If you want to change instead a system using speed then you can easily work with units like
$$ \begin{aligned} {\rm Time} & = \frac{\rm Length}{\rm Speed} 
\\ {\rm Acceleration} & =  \frac{\rm Speed^2}{\rm Length} 
\\ {\rm Force} & =  \frac{\rm Mass\; Speed^2}{\rm Length} 
\\ {\rm Energy} & =  {\rm Mass}\;{\rm Speed^2} 
\end{aligned}$$
but it won't change fundamentally the facts about the laws of motion and gravity.
A: There's two rational reasons why this would be a bad idea for everyday life:


*

*most speeds we encounter are significantly smaller than $c$. 

*it would change our units system entirely


For the first point, I'd be traveling down my street at 0.00000037 instead of 25 mph (40 km/h). Airplanes would travel at about 0.0000002 instead of about 600 mph (965 km/h).
For the second point, for velocity to be unitless, we'd need to have time and distance with the same units. Accelerations would then have units of 1/s, so that forces are measured as kg/s. And so on.
A: From the point of view of dimensional analysis alone, this can be done. It would not give mathematically inconsistent results. Mathematically, any quantities $A$, $B$, $C$ can be chosen as 'fundamental', as long as no relationship of the form $f(A, B, C) = 0$ exists between them.
However, from the point of view of physics itself, such an approach needs to be questioned. In physics, we consider certain concepts to be primitive - meaning that they cannot be explained in terms of simpler ideas. Space and time are amongst such ideas. Other concepts are derived - meaning that they can be explained or defined in terms of existing ideas. 
If we consider time and length as fundamental quantities, we should also be able to provide a simple prescription for measuring them. This can be easily done for time and length - for eg. - as the no. of oscillations of a particular pendulum, or as the no. of multiples of some unit length. In the case of velocity however, it would be quite difficult, or perhaps impossible to provide such a prescription which does not involve using the ideas of space or time. Hence, there does not appear to be a strong case for considering velocity as 'fundamental'. It does appear that certain quantities are really more fundamental than others, it does not appear to be just a conventional element.
A: Personally, I don't see why not. And I too believe it has the potential to open up new lines of inquiry and understanding. I have long thought the concepts of velocity and change have a good chance of in fact being most fundamental. As an example, what if the reality of the fundamental nature of our universe is such that, at an event, information is emitted at an angle, with the angle 0 being the minimum possible angle, so that the speed of light (c) is actually the minimum velocital angle 0. Slower speeds in our current way of thinking about such things would simply be an angle greater than 0 so that it would take longer, hence more "time", for that information to arrive at and influence some distant point in space.
