Chapter 2.3, Problem 27P

A body of constant mass $m$ is projected vertically upward with an initial velocity $v_0$ in a medium offering a resistance $k\left|v\right|$, where $k$ is a constant. Neglect changes in the gravitational force.

Find the maximum height $x_m$ attained by the body and the time $t_m$ at which this maximum height is reached.

Show that if $k{v}_0/mg<1$, then $t_m$ and $x_m$ can be expressed as

$t_m=\frac{v_0}{g}\left[1-\frac{1}{2}\frac{k{v}_0}{mg}+\frac{1}{3}{\left(\frac{k{v}_0}{mg}\right)}^2-\mathrm{...}\right]$’

$x_m=\frac{v_0^2}{2g}\left[1-\frac{2}{3}\frac{k{v}_0}{mg}+\frac{1}{2}{\left(\frac{k{v}_0}{mg}\right)}^2-\mathrm{...}\right]$.

Show that the quantity $k{v}_0/mg$ is dimensionless.