Question

The maximum safe speed for a banked road is intended to be increased by $20$%. If the angle of banking is not changed, then the radius of curvature of the road should be changed from $30\text{m}$ to $\text{m}$

Answer 1

Given $r_{old}=r=30\text{m}$. Let $r_{new}=r^{\prime }$
and $v_{new}=v^{\prime }$
Then, $v^{\prime }=v+\frac{20v}{100}=\frac{12v}{10}$
We know the condition of banking
$\tan \theta =\frac{v^2}{rg}$
$\therefore v^2=rg\tan \theta ....\left(i\right)$

Putting old and new values in Eq. ($i$) and then dividing will yield:
$\frac{v_{old}^2}{v_{new}^2}=\frac{v^2}{{v^{\prime }}^2}=\frac{rg\tan \theta }{r^{\prime }g\tan \theta }$
$\Rightarrow \frac{v}{v^{\prime }}=\sqrt{\frac{r}{r^{\prime }}}$
$\Rightarrow r^{\prime }=r\times \frac{{v^{\prime }}^2}{v^2}$
i.e $r^{\prime }=\frac{r\times {\left(\frac{12v}{10}\right)}^2}{v^2}=1.44r$
$\therefore r^{\prime }=1.44\times 30\text{m}=43.2\text{m}$