Question

$\huge\underline{QuEStiON}$

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Answer 1

$\huge\bf{\underline{\underline{\gray{GIVEN:-}}}}$

• The S.I unit of energy is $\bf\red{kg.m^2.s^{-2}}$ .

• The S.I unit of Speed is $\bf\red{m.s^{-1}}$ .

• The S.I unit of acceleration is $\bf\red{m.s^{-2}}$

• m stands for mass of the body .

$\huge\bf{\underline{\underline{\gray{TO\:FIND:-}}}}$

• The formulae for kinetic energy given in the question, rule out by dimensional arguments .

$\huge\bf{\underline{\underline{\gray{SOLUTION:-}}}}$

We know that,

$\huge\red\checkmark$ According to the principle of homogeneity of dimensional analysis, the dimensions of each term on both the sides of the correct formula or equation will be the same .

✞︎ The dimensional formula for K.E is $\bf\blue{[M^1\:L^2\:T^{-2}]}$ .

☯︎ According to the question,

• The dimensions of quantity on R.H.S of different equations are same, i.e. $\bf{[M^1\:L^2\:T^{-2}]}$ .

Now, we analyze all the five given options dimensionally .

(a) $\rm\bold{K\:=\:m^2\:v^3}$

Where,

• $\bf\red{m}$ = Mass

• $\bf\red{v}$ = Velocity

$\rm{:\implies\:K\:=\:[M]^2\:[L^1\:T^{-1}]^3\:}$

$\bf\purple{:\implies\:K\:=\:[M^2\:L^3\:T^{-3}]\:}$

$\huge\red\therefore$ This is not clearly match with the dimensional formula of K.E .

(b) $\rm\bold{K\:=\:(1/2)\:m\:v^2}$

[NOTE → Constant is a dimension less quantity .]

$\rm{:\implies\:K\:=\:[M]\:[L^1\:T^{-1}]^2\:}$

$\bf\purple{:\implies\:K\:=\:[M^1\:L^2\:T^{-2}]\:}$

$\huge\red\therefore$ This is obviously correct match with the dimensional formula of K.E .

(c) $\rm\bold{K\:=\:m\:a}$

Where,

• $\bf\red{m}$ = Acceleration

$\rm{:\implies\:K\:=\:[M]\:[L^1\:T^{-2}]\:}$

$\bf\purple{:\implies\:K\:=\:[M^1\:L^1\:T^{-2}]\:}$

$\huge\red\therefore$ This is not clearly match with the dimensional formula of K.E .

(d) $\rm\bold{K\:=\:(3/16)\:m\:v^2}$

[NOTE → Constant is a dimension less quantity .]

$\rm{:\implies\:K\:=\:[M]\:[L^1\:T^{-1}]^2\:}$

$\bf\purple{:\implies\:K\:=\:[M^1\:L^2\:T^{-2}]\:}$

$\huge\red\therefore$ This is obviously correct match with the dimensional formula of K.E .

(e) $\rm\bold{K\:=\:(1/2)\:m\:v^2\:+\:m\:a}$

$\rm{:\implies\:K\:=\:[M]\:[L^1\:T^{-1}]^2\:+\:[M]\:[L^1\:T^{-2}]\:}$

$\bf\purple{:\implies\:K\:=\:[M^1\:L^2\:T^{-2}]\:+\:[M^1\:L^1\:T^{-2}]\:}$

NOTE → Here first term in the R.H.S is "(1/2) mv²" which will give the dimension of K.E, but second term in the R.H.S is "ma" which will give the dimension of force .

$\huge\red\therefore$ To make the entire expression is dimensionally incorrect .

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$\huge\blue\therefore$ The correct options to be chosen here are (a), (c) & (e) as all these can be ruled out based on dimensional arguments .