A student says: "I can find s = ut + 1/2at² by dimensions alone." Explain why not, and give two…

Step 1. The sum. The method gives one product, $k\,u^{x}a^{y}t^{z}$. The formula is a sum of two terms, $ut$ and $\frac{1}{2}at^2$, so it cannot come out.

Step 2. Pure numbers. It cannot find the $\frac{1}{2}$, or a $2\pi$ anywhere.

Step 3. More limits. It cannot give a relation with $\sin$, $\log$ or $e$ in it; it fails when more quantities are involved than there are base dimensions; and it cannot tell work from torque.