## College Physics (4th Edition)

(a) $v = \sqrt{\frac{B}{\rho}}$ The equation gives the speed of sound in units of m/s (b) $v = C\sqrt{\frac{B}{\rho}}$ No other combination of $B$ and $\rho$ can give the dimensions of speed, so the given equation must be correct.
(a) Speed $v$ is measured in units of $m~s^{-1}$ $B$ is measured in units of $N/m^2$ which can be expressed as $kg~m^{-1}~s^{-2}$ $\rho$ is measured in units of $kg~m^{-3}$ $v = \sqrt{\frac{B}{\rho}}$ We can consider the units: $\sqrt{\frac{N/m^2}{kg/m^3}} = \sqrt{\frac{kg/m~s^2}{kg/m^3}} = \sqrt{\frac{m^2}{s^2}} = m/s$ The equation gives the speed of sound in units of m/s (b) Let's assume that $v = C~B^a~\rho^b$, where $C$ is a dimensionless constant. Then: $m~s^{-1} = (kg~m^{-1}~s^{-2})^a~(kg~m^{-3})^b$ We can consider the units of $s$: $(s^{-2})^a = s^{-1}$ $s^{-2a} = s^{-1}$ $-2a = -1$ $a = \frac{1}{2}$ We can consider the units of $m$: $(m^{-1})^a~(m^{-3})^b = m^1$ $-a-3b = 1$ $3b = -a-1$ $3b = -\frac{1}{2}-1$ $3b = -\frac{3}{2}$ $b = -\frac{1}{2}$ We can use the exponents to write the equation: $v = C~B^a~\rho^b$ $v = C~B^{1/2}~\rho^{-1/2}$ $v = C\sqrt{\frac{B}{\rho}}$ No other combination of $B$ and $\rho$ can give the dimensions of speed, so the given equation must be correct.