Question

A physical quantity $A$ on multiplication with velocity results in another quantity $B$. If the quantity $B$ is energy, then the quantity $A$ is

• A. mass
• B. momentum
• C. force
• D. acceleration
• E. power

Answer 1

Answer: (B)

Given, $A\times v=B$
where, $B$ is energy.
$\Rightarrow A\times v=$ " Energy "
Dimensional formula of $v,[v]=\left[L{T}^{-1}\right]$
Dimensional formula of $E$,
$[E]=\left[M{L}^2{T}^{-2}\right]$
Let dimensional formula of quantity $A=\left[M^a{L}^b{T}^c\right]$
$\Rightarrow \left[M^a{L}^b{T}^c\right]\left[L{T}^{-1}\right]=\left[M{L}^2{T}^{-2}\right]$
$\Rightarrow \left[M^a{L}^{b+1}{T}^{c-1}\right]=\left[M{L}^2{T}^{-2}\right]$
From the principle of homogeneity,
$a=1$
$b+1=2\Rightarrow b=1$
$c-1=-2\Rightarrow c=-1$
$\Rightarrow \text{ Dimensional formula of }A=\left[M{L}^0{T}^{-1}\right]$
$=\left[ML{T}^{-1}\right]$
This is the dimensional formula of momentum.