Dear Uncle Colin,

I can derive the timeless SUVAT equation $v^2 = u^2 + 2as$, but I can’t intuitively see where it comes from. Any clues?

- Everyone Needs Explanations, Really Get Yours

Hi, ENERGY, and thanks for your message!

This is one that I never really picked up on for a long time - but when I finally saw it, it made perfect sense.

The key trick: multiply the whole thing by $\frac{1}{2}m$.

You then have $\frac{1}{2}mv^2 = \frac{1}{2}mu^2 + (ma)s$.

Now, $\frac{1}{2}mv^2$ is the kinetic energy when you reach the final velocity, and $\frac{1}{2}mu^2$ the kinetic energy as you start.

But energy is conserved, right? Aha! Unless there’s a force acting on the object. If there’s a force acting on the object, then there may be work being done: in particular - keeping things to one dimension - if a force $F$ acts over a displacement $s$, it adds $Fs$ to the kinetic energy. (This is the work-energy principle).

And, since Newton 2 tells us $F=ma$, the timeless SUVAT equation is simply a rearrangement of this.

Hope that helps!

- Uncle Colin