ऊर्जा का $SI$ मात्रक $J = kg\, m ^{2} s ^{-2}$ है, चाल $v$ का $m s ^{-1}$ और त्वरण $a$ का $m s ^{-2}$ है। गतिज ऊर्जा $(k)$ के लिए निम्नलिखित सूत्रों में आप किस-किस को विमीय दृष्टि से गलत बताएँगे ? $(m$ पिण्ड का द्रव्यमान है )।

$(a)$ $K=m^{2} v^{3}$

$(b)$ $K=(1 / 2) m v^{2}$

$(c)$ $K=m a$

$(d)$ $K=(3 / 16) m w^{2}$

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

Answer Every correct formula or equation must have the same dimensions on both sides of the equation. Also, only quantities with the same physical dimensions can be added or subtracted. The dimensions of the quantity on the right side are $\left[ M ^{2} L ^{3} T ^{-3}\right]$ for $( a ) ; \left[ M L ^{2} T ^{-2}\right]$ for $(b)$ and $(d)$: $\left[ MLT ^{-2}\right]$ for $(c)$. The quantity on the right side of $(e)$ has no proper dimensions since two quantities of different dimensions have been added. since the kinetic energy $K$ has the dimensions of $\left[ M L ^{2} T ^{-2}\right],$ formulas $(a), (c)$ and $(e)$ are ruled out. Note that dimensional arguments cannot tell which of the two, $(b)$ or $(d)$, is the correct formula. For this, one must turn to the actual definition of kinetic energy . The correct formula for kinetic energy is given by $(b)$.