The position of a body with acceleration '$a$' is given by $x = K{a^m}{t^n}$,where $t$ is time. Find the dimensions of $m$ and $n$.

If mass is written as $m=kc^{p} G^{-1 / 2} \,h^{1 / 2}$ then the value of $P$ will be : (Constants have their usual meaning with $k$ a dimensionless constant)

The equation $\frac{dV}{dt} = At - BV$ describes the rate of change of velocity of a body falling from rest in a resisting medium. The dimensions of $A$ and $B$ are

The $SI$ unit of energy is $J = kg \, m^{2} \, s^{-2}$; that of speed $v$ is $m \, s^{-1}$ and of acceleration $a$ is $m \, s^{-2}$. Which of the formulae for kinetic energy $(K)$ given below can you rule out on the basis of dimensional arguments ($m$ stands for the mass of the body):
$(a)$ $K = m^{2} v^{3}$
$(b)$ $K = (1/2) m v^{2}$
$(c)$ $K = m a$
$(d)$ $K = (3/16) m v^{2}$
$(e)$ $K = (1/2) m v^{2} + m a$

If momentum $(P)$,area $(A)$,and time $(T)$ are taken to be fundamental quantities,then energy has the dimensional formula of:

If Surface tension $(S)$,Moment of Inertia $(I)$ and Planck's constant $(h)$ were to be taken as the fundamental units,the dimensional formula for linear momentum would be