Consider the equation $\frac{1}{2} m v^{2} = m g h$,where $m$ is the mass of the body,$v$ is the velocity,$g$ is the acceleration due to gravity,and $h$ is the height. Is this equation dimensionally correct?

If the unit of force is $1 \,kN$,the unit of length is $1 \,km$,and the unit of time is $100 \,s$,what will be the unit of mass in $kg$?

If $E, M, J$ and $G$ respectively denote energy, mass, angular momentum, and universal gravitational constant, the quantity which has the same dimensions as the dimensions of $\frac{E J^2}{M^5 G^2}$ is:

The expressions below give current $I$ through an electronic component as a function of applied potential $V$. $I_0$ and $V_0$ are constants having dimensions of current and potential respectively. Which of the following are dimensionally incorrect?
$(A)$ $I=I_0\left(e^{\frac{2 V}{V_0}}+1\right)$
$(B)$ $I=I_0\left(e^{\frac{V}{2 V_0}}-1\right)$
$(C)$ $I=I_0 V_0\left(e^{\frac{V}{V_0}}-1\right)$
$(D)$ $I=I_0\left(\frac{V}{V_0}\right)\left(e^{\frac{V}{V_0}}-1\right)$

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.

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: