Question

The SI unit of energy has been mentioned as $J=kg{{m}^{2}}{{s}^{-2}}$. The SI unit of speed $v$ is $m{{s}^{-1}}$ and the SI unit of acceleration $a$ is $m{{s}^{-2}}$. Which of the formula given in the question for the kinetic energy $\left( K \right)$ mentioned below can you rule out on the basis of dimensional arguments. Here we can say that $m$ stands for the mass of the body. This question may contain multiple answers.
$\begin{align}
  & A.K={{m}^{2}}{{v}^{3}} \\
 & B.K=\dfrac{1}{2}m{{v}^{2}} \\
 & C.K=ma \\
 & D.K=\left( \dfrac{3}{16} \right)m{{v}^{2}} \\
 & E.K=\dfrac{1}{2}m{{v}^{2}}+ma \\
\end{align}$

Answer 1

Hint: Compare the right hand side and the left hand side of each of the equations. According to the principle of homogeneity of dimensional analysis, the dimensions of each term on either the sides of the correct equation or the formula will be identical to each other. This will help you in answering this question.

Complete answer:
The dimensional formula for K.E is found to be given as,
$\left[ M{{L}^{2}}{{T}^{-2}} \right]$
According to the principle of homogeneity of dimensional analysis, the dimensions of each term on either the sides of the correct equation or the formula will be identical to each other. Therefore the dimensions of quantity on right hand side of various quantities are given as,
$\begin{align}
  & 1.\left[ {{M}^{2}}{{L}^{3}}{{T}^{-3}} \right] \\
 & 2.\left[ M{{L}^{2}}{{T}^{-2}} \right] \\
 & 3.\left[ {{M}^{1}}{{L}^{1}}{{T}^{-2}} \right] \\
 & 4.\left[ M{{L}^{2}}{{T}^{-2}} \right] \\
 & 5.\left[ M{{L}^{2}}{{T}^{-2}} \right]+\left[ ML{{T}^{-2}} \right] \\
\end{align}$
As we can see that the dimensional formula of the right hand side of the equations mentioned in the option A, C and both the terms of the right hand side of the equation E are not found to be identical as that of the left hand side. That is we can write that,
$\left[ M{{L}^{2}}{{T}^{-2}} \right]$
Therefore among the equations given. We can rule out option A, C, and E dimensionally as they are not matching.

Hence the answers for this question will be option B and D.

Note:
Dimensional analysis can be otherwise called as the factor-label method or the unit-factor method. This is a method used in order to convert one unit into a different unit. In order to do this, we will make use of a conversion factor, which will be a numerical quantity which will be multiplied or divided to the quantity or number that we need to convert.