Given the expression for the escape velocity $v_e = \sqrt{\frac{2GM}{R}}$, where $G$ is the gravitational constant, $M$ is mass, and $R$ is radius. If a new quantity $Y$ is defined as $Y = \frac{v_e^3}{\sqrt{GM}}$, what are the dimensions of $Y$?

If $S = \frac{1}{3}f{t^3},f$ has the dimensions of

The equation of state of some gases can be expressed as $\left( {P + \frac{a}{{{V^2}}}} \right)\left( {V - b} \right) = \;RT$. Here P is the pressure, V is the volume, T is the absolute temperature and a,b,R are constants. The dimensions of ‘a’ are

If the product of energy ($E$) and a quantity $X$ has the dimensions of $G$ (the gravitational constant), what are the dimensions of $X$?

The velocity of a particle (v) at an instant t is given by $v = at + b{t^2}$ the dimension of b is

Given that $:y = A\sin \left[ {\left( {\frac{{2\pi }}{\lambda }} \right)\;\left( {ct - x} \right)} \right]$ where, y and x are measured in metre. Which of the following statements is true?

The period of a body under SHM is represented by $T = {P^a}{D^b}{S^c}$; where P is pressure,

D is density and S is surface tension. The value of a,b and c are

In the relation $p = \frac{{\rm{\alpha }}}{{\rm{\beta }}}{\rm{\;}}{{\rm{e}}^{ - \frac{{\alpha z}}{{k{\rm{heta }}}}}},{\rm{\;}}p$ , is the pressure, z the distance, k is Boltzmann constant and $heta$ is the temperature, the dimensional formula of $\beta$ will be

The position of a particle at time t is given by the relation $x\left( t \right) = \left( {\begin{array}{*{20}{c}}

{\frac{{{v_0}}}{\alpha }}\

;

\end{array}} \right)\left( {1 - {e^{\alpha t}}} \right),;$where${v_0}$is constant and$\alpha > 0$. The dimensions of${v_0}$and$\alpha $ are respectively

If the dimensions of a physical quantity are $ML^2T^{-2}$, what could it represent?

The speed of sound (v) in a gas depends on its pressure (P) and density ($\rho$). Using dimensional analysis, determine the relationship between v, P, and $\rho$.