Question

In the following $x$ is in meters, $t$ in seconds, $v$ in $\text{m/s}$ and $a$ is in ${\text{m/s}}^2$. Find the SI unit of each combination.

• A. $\frac{v^2}{xa}$ is dimension less
• B. $\sqrt{\frac{x}{a}}$ has dimensions same as time
• C. $\frac{1}{2}a{t}^2$ is unitless
• D. A dimensionless physical quantity may have an unit

Answer 1

Answer: (A), (B), (D)

The correct options are
A $\frac{v^2}{xa}$ is dimension less
B $\sqrt{\frac{x}{a}}$ has dimensions same as time
D A dimensionless physical quantity may have an unit

(a) $\frac{v^2}{xa}$
$=\frac{(\text{m/s}{)}^2}{{\text{mm/s}}^2}=\frac{(\text{m/s}{)}^2}{{\text{(m/s)}}^2}=$ dimensionless
(b) $\sqrt{\frac{\text{m}}{{\text{m/s}}^2}}=\sqrt{{\text{s}}^2}=s$
(c) Here constant $\frac{1}{2}$ does not have any units
but $a{t}^2$ is displacement.
(d) Eg.- angle