According to the Principle of Homogeneity of dimensions, the magnitudes of physical quantities may be added together or subtracted from one another only if they have the same dimensions .

1. In the given equation $S = a + bt + ct^2$, the quantity $S$ represents a length (measured in metres).
2. Therefore, each term on the right-hand side ($a$, $bt$, and $ct^2$) must also have the dimensions of length $[L]$ and the unit of metres ($m$).
3. For the term $ct^2$: $\text{Unit of } (ct^2) = \text{Unit of } S$ $(\text{Unit of } c) \times (\text{Unit of } t)^2 = m$ $(\text{Unit of } c) \times s^2 = m$ $\text{Unit of } c = m/s^2 = ms^{-2}$

This is consistent with the standard kinematic equation $x = x_0 + v_0t + \frac{1}{2}at^2$, where the coefficient of $t^2$ involves acceleration ($a$) with units $ms^{-2}$.