The relationship of the linear speed vv to the angular speed ω (in radians per unit time) and radius r is:

\(\displaystyle{v}=ω{r}\)

Convert the angular speed to radians per hour:

\(\displaystyleω={\left({13}{r}{e}\frac{{v}}{\min}\right)}\cdot{\left({2}π{r}{a}\frac{{d}}{{1}}{r}{e}{v}\right)}\cdot{\left({60}\frac{\min}{{1}}{h}{r}\right)}={1560}π{r}{a}\frac{{d}}{{h}}{r}\)

Convert the radius to miles:

\(\displaystyle{r}={25}{f}{t}\cdot{\left({1}{m}\frac{{i}}{{5280}}{f}{t}\right)}=\frac{{25}}{{5280}}{m}{i}\le{s}\)

So, the linear speet is:

\(\displaystyle{v}={1560}π{r}{a}\frac{{d}}{{h}}{r}\cdot{\left(\frac{{25}}{{5280}}\right)}{m}{i}\le{s}\)

\(\displaystyle{v}≈{23.2}{m}\frac{{i}}{{h}}{r}\)

\(\displaystyle{v}=ω{r}\)

Convert the angular speed to radians per hour:

\(\displaystyleω={\left({13}{r}{e}\frac{{v}}{\min}\right)}\cdot{\left({2}π{r}{a}\frac{{d}}{{1}}{r}{e}{v}\right)}\cdot{\left({60}\frac{\min}{{1}}{h}{r}\right)}={1560}π{r}{a}\frac{{d}}{{h}}{r}\)

Convert the radius to miles:

\(\displaystyle{r}={25}{f}{t}\cdot{\left({1}{m}\frac{{i}}{{5280}}{f}{t}\right)}=\frac{{25}}{{5280}}{m}{i}\le{s}\)

So, the linear speet is:

\(\displaystyle{v}={1560}π{r}{a}\frac{{d}}{{h}}{r}\cdot{\left(\frac{{25}}{{5280}}\right)}{m}{i}\le{s}\)

\(\displaystyle{v}≈{23.2}{m}\frac{{i}}{{h}}{r}\)