Question

Dimensions of kinetic energy are :

A. $[M{{L}^{2}}{{T}^{-2}}]$
B. $[{{M}^{2}}{{L}^{{}}}{{T}^{-1}}]$
C. $[M{{L}^{2}}{{T}^{-1}}]$
D. $[M{{L}^{3}}{{T}^{-1}}]$

Answer 1

We are supposed to find the dimensional formula of kinetic energy. For that, we have to analyse the definition and the numerical formula of the same. Further, we can deduce the dimensional formula by finding the degree of dependence of a physical quantity on another. The principle of consistency of two expressions can be used to find the equation relating these two quantities.

Formulas used:
$KE=\dfrac{1}{2}m{{v}^{2}}$, where $KE$ is the kinetic energy, $m$ is the mass and $v$ is the velocity.

Complete step by step answer:
We know that the value of kinetic energy is obtained from the formula
$KE=\dfrac{1}{2}m{{v}^{2}}$
The SI unit of mass is kg and that of velocity is m/s

Hence, $KE=\dfrac{1}{2}kg\dfrac{{{m}^{2}}}{{{s}^{2}}}$
On neglecting the constant, we get the unit of Kinetic Energy as
$\begin{aligned} &\Rightarrow kg{{m}^{2}}{{s}^{-2}} \\\ \end{aligned}$
$[KE]=\dfrac{[kg]{{[m]}^{2}}}{{{[s]}^{2}}}$
The dimensional formula for mass is $[M]$
The dimensional formula for length is $[L]$
The dimensional formula for time is $[T]$
$[KE]=\dfrac{[M]{{[L]}^{2}}}{{{[T]}^{2}}}=[M{{L}^{2}}{{T}^{-2}}]$
Therefore, we can represent the dimensional formula of kinetic energy as $[M{{L}^{2}}{{T}^{-2}}]$
Therefore, option A is the correct choice among the four.

Note: Dimensional formula is widely used in many areas. However, there are a few problems along the way. Dimensionless quantities and proportionality constant cannot be determined in this way. It does not apply to trigonometric, logarithmic and exponential functions. When we look at a quantity that is dependent on more than three quantities, this approach will be difficult. In line with all of this, if one side of our equation has addition or subtraction of quantities, this approach is not appropriate.