If x =a + bt + ct², where x is in metre and t in second; then what is the unit of c ?

The correct option is ms⁻².

To find the unit of the constant $c$, we can apply the Principle of Homogeneity of Dimensions. According to this principle, terms that are added, subtracted, or equated in a physical equation must have the same dimensions and units.

The given equation is:
$x=a+bt+c{t}^2$
where:
- $x$ is the distance, measured in metres (m).
- $t$ is the time, measured in seconds (s).

By the principle of homogeneity, the unit of each individual term on the right-hand side ($a$, $bt$, and $c{t}^2$) must be equal to the unit of the term on the left-hand side ($x$).

Therefore, the unit of the term $c{t}^2$ is equal to the unit of $x$:
$\text{Unit of}\left(c{t}^2\right)=\text{Unit of}x$
$\left(\text{Unit of}c\right)\times (\text{Unit of}t{)}^2=\text{Unit of}x$

Now, we isolate the unit of $c$:
$\text{Unit of}c=\frac{\text{Unit of}x}{{\left(\text{Unit of}t\right)}^2}$

Substituting the units of distance (metres, m) and time (seconds, s) into the relation:
$\text{Unit of}c=\frac{m}{s^2}=m{s}^{-2}$

Hence, the unit of $c$ is ms^-2.