Torricelli's Equation ($v^2=u^2+2as$) is usually presented as the particular formulation of the SUVAT system which doesn't involve t. It is derived from the others using some (perhaps well-motivated) algebraic tricks. Students are then advised to use it when they know three of $s,u,v$ and $a$, but not $t$.
Can anyone provide physical insight into this equation, how it is derived and in what situations is it useful.
It's just conservation of energy without being called as such.
$$\mbox{Energy}_{\mbox{before}}=\mbox{Energy}_{\mbox{after}}$$
$$\frac{1}{2}m v_1^2+m g h_1=\frac{1}{2}m v_2^2 +m g h_2$$
$$\frac{1}{2}m v_1^2=\frac{1}{2}m v_2^2 +m g (h_2-h_1)$$
$$v_1^2=v_2^2 +2g \Delta y$$
This seems trivial when you have calculus and a concept of energy, but without calculus it makes it seem like an important equation derived from the formula $y(t)=y_0+v_0t-\frac{1}{2}g t^2$.