Which of the following relations can be derived using dimensional analysis?

$A$ bomb explodes at time $t=0$ in a uniform,isotropic medium of density $\rho$ and releases energy $E$,generating a spherical blast wave. The radius $R$ of this blast wave varies with time $t$ as

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):
$(a)$ $K = m^{2} v^{3}$
$(b)$ $K = (1/2) m v^{2}$
$(c)$ $K = m a$
$(d)$ $K = (3/16) m v^{2}$
$(e)$ $K = (1/2) m v^{2} + m a$

The energy $E$ of a system is a function of time $t$ and is given by $E(t) = \alpha t - \beta t^3$,where $\alpha$ and $\beta$ are constants. The dimensions of $\alpha$ and $\beta$ are

If energy is given by $U = \frac{A\sqrt{x}}{x^2 + B}$,then the dimensions of $AB$ are:

If dimensions of critical velocity $v_c$ of a liquid flowing through a tube are expressed as $[\eta^x \rho^y r^z]$,where $\eta, \rho,$ and $r$ are the coefficient of viscosity of the liquid,density of the liquid,and radius of the tube respectively,then the values of $x, y,$ and $z$ are given by: