Let us consider an equation

$\frac{1}{2} m v^{2}=m g h$

where $m$ is the mass of the body. velocity, $g$ is the acceleration do gravity and $h$ is the height. whether this equation is dimensionally correct. 

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Answer The dimensions of $LHS$ are

$[ M ]\left[ L T ^{-1}\right]^{2}=[ M ]\left[ L ^{2} T ^{-2}\right]$

$=\left[ M L ^{2} T ^{-2}\right]$

The dimensions of $RHS$ are

$[ M ]\left[ L T ^{-2}\right][ L ]=[ M ]\left[ L ^{2} T ^{-2}\right]$

$=\left[ M L ^{2} T ^{-2}\right]$

The dimensions of $LHS$ and $RHS$ are the same and hence the equation is dimensionally correct.

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