Given that $v$ is speed, $r$ is the radius and $g$ is the acceleration due to gravity. Which of the following is dimensionless

In a typical combustion engine the work done by a gas molecule is given $W =\alpha^{2} \beta e ^{\frac{-\beta x ^{2}}{ KT }}$, where $x$ is the displacement, $k$ is the Boltzmann constant and $T$ is the temperature. If $\alpha$ and $\beta$ are constants, dimensions of $\alpha$ will be

The velocity of a freely falling body changes as ${g^p}{h^q}$ where g is acceleration due to gravity and $h$ is the height. The values of $p$ and $q$ are

The velocity $v$ (in $cm/\sec $) of a particle is given in terms of time $t$ (in sec) by the relation $v = at + \frac{b}{{t + c}}$ ; the dimensions of $a,\,b$ and $c$ are

If the buoyant force $F$ acting on an object depends on its volume $V$ immersed in a liquid, the density $\rho$ of the liquid and the acceleration due to gravity $g$. The correct expression for $F$ can be