Question

Position of a body with acceleration ${}^{\prime }{a}^{\prime }$ is given by $x=k{a}^m{t}^n$, here $t$ is time. Find dimension of $m$ and $n$.

• A. $m=2,n=1$
• B. $m=1,n=1$
• C. $m=2,n=2$
• D. $m=1,n=2$

Answer 1

The correct option is D $m=1,n=2$

$x=K{a}^m{t}^n\dots \left(i\right)$

Where,

$K$=constant

Dimension of $a=\left[L{T}^{-2}\right]$

Dimension of $x=\left[L\right]$

Dimension of $t=\left[T\right]$

By putting all dimensions in equation (i) we get

$\left[M^0{L}^1{T}^0\right]=\left[M^0{L}^1{T}^{-2}{]}^m\right[M^0{L}^0{T}^1{]}^n$

Comparing the quantities we get,

$m=1\dots \left(ii\right)$

$-2m+n=0\dots \left(iii\right)$

By putting the value of $m=1$ in equation $\left(iii\right)$ we get,

$-2\left(1\right)+n=0$

$n=2$

Dimension of $m$ & $n$ are $1,2$