If x = at + bt² , where x is the distance travelled by the body in kilometres while t is the time in seconds, then the unit of b is

The correct option is km/s².

To find the unit of the constant
$b$
in the given equation, we use the principle of dimensional homogeneity. According to this principle, the terms on both sides of a physical equation must have the same dimensions and units.

The given equation is:
$x=at+b{t}^2$
where:
-
$x$
is the distance in kilometres (km).
-
$t$
is the time in seconds (s).

According to the principle of homogeneity, the unit of each term on the right-hand side must be equal to the unit of the term on the left-hand side. Therefore, the unit of the term
$b{t}^2$
must be the same as the unit of
$x$
:

$\text{Unit of}\left(b{t}^2\right)=\text{Unit of}x$

We can separate the product of units as:
$\left(\text{Unit of}b\right)\times {\left(\text{Unit of}t\right)}^2=\text{Unit of}x$

Substituting the given units of distance (km) and time (s):
$\left(\text{Unit of}b\right)\times {\text{s}}^2=\text{km}$

Solving for the unit of
$b$
by isolating it on one side:
$\text{Unit of}b=\frac{\text{km}}{{\text{s}}^2}$

Thus, the unit of
$b$
is km/s².