Question

7.5 air resistance and acceleration: when modeling the effect of resistive forces for an object moving through fluids, typically the forces are approximated as proportional to the objects velocity, v, and velocity squared, v² (the relative strength of these terms depends on the objects size, speed, and fluid properties). if the magnitude of the acceleration due to the linear and quadratic drag forces are given by (a_{lin}=\frac{b}{m}v) and (a_{quad}=\frac{c}{m}v^{2}), where m is the objects mass, what are the units of b and c?

Explanation:

Step1: Recall the unit of acceleration

The unit of acceleration $a$ is $\frac{m}{s^{2}}$, the unit of mass $m$ is $kg$, and the unit of velocity $v$ is $\frac{m}{s}$.

Step2: Find the unit of $b$ from the linear - drag formula

We have $a_{lin}=\frac{b}{m}v$. Rearranging for $b$ gives $b = \frac{a_{lin}m}{v}$. Substituting the units: $a_{lin}$ has units $\frac{m}{s^{2}}$, $m$ has units $kg$, and $v$ has units $\frac{m}{s}$. Then $b$ has units $\frac{\frac{m}{s^{2}}\times kg}{\frac{m}{s}}=kg/s$.

Step3: Find the unit of $c$ from the quadratic - drag formula

We have $a_{quad}=\frac{c}{m}v^{2}$. Rearranging for $c$ gives $c=\frac{a_{quad}m}{v^{2}}$. Substituting the units: $a_{quad}$ has units $\frac{m}{s^{2}}$, $m$ has units $kg$, and $v$ has units $\frac{m}{s}$. Then $c$ has units $\frac{\frac{m}{s^{2}}\times kg}{(\frac{m}{s})^{2}}=kg/m$.

Answer:

The unit of $b$ is $kg/s$ and the unit of $c$ is $kg/m$.