Consider two physical quantities $A$ and $B$ related to each other as $E = \frac{B - x^2}{At}$,where $E, x,$ and $t$ have dimensions of energy,length,and time respectively. The dimension of $AB$ is

In the equation,pressure $P = \frac{c - t^{2}}{DS}$,where $S$ and $t$ represent the distance and time respectively. The dimensions of $\left(\frac{D}{c}\right)$ are

Assume that the drag force on a football depends only on the density of the air,the velocity of the ball,and the cross-sectional area of the ball. Balls of different sizes but the same density are dropped in an air column. The terminal velocity reached by balls of masses $250 \,g$ and $125 \,g$ are in the ratio:

The dimensional formula of $\frac{1}{\mu_{0} \varepsilon_{0}}$ is . . . . . . .

Consider an expression $Q V = k P T L^\alpha$ where $V, P, T, L$ are volume, pressure, time, and length respectively. The quantity $[Q]$ has dimension $M L^{-1} T^{-1}$. $k$ is a dimensionless constant. The value of the integer $\alpha$ is:

The speed of a water wave $v$ depends on its wavelength $\lambda$,the density of water $\rho$,and the acceleration due to gravity $g$. Determine the relationship between these quantities.