Which of the following formulae for kinetic energy ($K$) can be ruled out on the basis of dimensiona

Question

Which of the following formulae for kinetic energy ($K$) can be ruled out on the basis of dimensional arguments? ($m$ = mass, $v$ = speed, $a$ = acceleration) I. $K = m^2 v^3$ II. $K = \frac{1}{2}mv^2$ III. $K = ma$ IV. $K = \frac{3}{16}mv^2$ V. $K = \frac{1}{2}mv^2 + ma$

Professor PC Thomas Classes · Answer

Kinetic energy has dimensions $[M L^2 T^{-2}]$. Formula I gives $[M^2 L^3 T^{-3}]$. Formula III gives $[M L T^{-2}]$. Formula V adds two terms with different dimensions ($[M L^2 T^{-2}]$ and $[M L T^{-2}]$), which violates the principle of homogeneity. Thus, I, III, and V are dimensionally incorrect.