Which of the following pairs of units are used for measuring the same physical quantity ?

The correct option is kg m²/s², Ws.

To determine which pair of units measures the same physical quantity, let us analyze the dimensions and base units for each pair given in the option:

1. First unit: $\text{kg }{\text{m}}^2/{\text{s}}^2$

Recall the formula for kinetic energy:

$E=\frac{1}{2}m{v}^2$

Here, mass ($m$) is measured in kilograms ($\text{kg}$) and velocity ($v$) is measured in meters per second ($\text{m/s}$).

Therefore, the SI unit of energy is:

$\text{kg}\times {\left(\text{m/s}\right)}^2=\text{kg }{\text{m}}^2/{\text{s}}^2$

Thus, $\text{kg }{\text{m}}^2/{\text{s}}^2$ is a unit of Energy (or Work).

2. Second unit: $\text{Ws}$ (Watt-second)

Recall the relationship between energy, power, and time:

$\text{Energy}=\text{Power}\times \text{Time}$

Since Power is measured in Watts ($\text{W}$) and Time is measured in seconds ($\text{s}$):

$1\text{ Watt-second (Ws)}=1\text{ Joule (J)}$

Thus, $\text{Ws}$ is also a unit of Energy.

Since both $\text{kg }{\text{m}}^2/{\text{s}}^2$ and $\text{Ws}$ represent the physical quantity of Energy (or Work), they form the pair of units used for measuring the same physical quantity.