Assertion : The given equation $x=x_{0}+u_{0} t+\frac{1}{2} a t^{2}$ is dimensionally correct, where

Question

Assertion : The given equation $x=x_{0}+u_{0} t+\frac{1}{2} a t^{2}$ is dimensionally correct, where $x$ is the distance travelled by a particle in time $t$, initial position $x_{0}$, initial velocity $u_{0}$ and uniform acceleration $a$ is along the direction of motion.\ Reason : Dimensional analysis can be used for checking the dimensional consistency or homogeneity of the equation.

Professor PC Thomas Classes · Answer

(a) : Given equation: $x=x_0+u_0 t+\frac{1}{2} a t^2$. The dimensions of each term may be written as \ $ \begin{aligned} & {[x]=[\mathrm{L}] ;\left[x_0ight]=[\mathrm{L}] ;\left[u_0 tight]=\left[\mathrm{LT}^{-1}ight][\mathrm{T}]=[\mathrm{L}]} \ & {\left[\left(\frac{1}{2}ight) a t^2ight]=\left[\mathrm{LT}^{-2}ight]\left[\mathrm{T}^2ight]=[\mathrm{L}]} \end{aligned} \ $ As each term on the right hand side of this equation has the same dimension as left hand side of the equation, hence this equation is dimensionally correct.