The Tau of SUVAT

The Mathematical Ninja twiddled absent-mindedly with the nunchaku they used as a stress toy.

“Do you use SUVAT?” asked the student, timidly.

The Mathematical Ninja shrugged. “Yeah, I suppose that’s one way.”

He giggled. “I don’t know the SUVAT equations, though… I’ve been meaning to learn them.”

“I wouldn’t bother,” said the Mathematical Ninja, “you can work them out on the fly.”

“Is that easier?”

Another shrug. “You know about velocity-time graphs, right?”

The student nodded, enthusiastically. “The area underneath is the distance! And the gradient is the… acceleration.”

“Good. So you agree with me that $a=\frac{dv}{dt}$ at any given instant, and $s={\int }_0^{t}vd\tau$?”

“Why $\tau$?”

“Just good manners ¹,” said the Mathematical Ninja. “Don’t worry about it. So, if $a=\frac{dv}{dt}$, that means $[v{]}_0^t={\int }_0^{t}ad\tau$ - and if $a$ is a constant, that means $v-u=at$.”

“Again with the $\tau$! I suppose $u$ is just $v$ evaluated at $t=0$. But that sounds sort of familiar… $v=u+at$?”

“If you like,” said the Ninja. “It’s all the same.”

“Oo, I have an idea!” said the student. If $s={\int }_0^{t}vd\tau$, we can substitute in for $v$!”

Nod.

“So… $s={\int }_0^t(u+at)d\tau$?”

“A $\tau$ in the bracket, but otherwise fine.”

“Dagnabbit. Let’s see… $s={[u\tau +\frac{1}{2}a{\tau }^2]}_0^t$, which is… $ut+\frac{1}{2}a{t}^2$! It works! But aren’t there five equations?”

“There are,” said the Mathematical Ninja, “although the other two need a little thought. There’s area of a trapezium, which gives you $\frac{u+v}{2}t=s$.”

“… or $u+v=\frac{2s}{t}$… and you can use difference of two squares to get $(u+v)(v-u)=\frac{2s}{t}\times at$ - which means $v^2-u^2=2as$!”

The Mathematical Ninja looked, begrudgingly, impressed.

* Edited 2021-06-16 to correct TMN’s gender.

Footnotes:

1. Technically: you don’t want to have two $t$s meaning different things, so the limit needs to be a $t$ and the integration variable something else - $\tau$.