Question

A curved road of $50\text{m}$ radius is banked at correct angle for a given maximum speed $vm/s$. If the radius of curvature of the road is changed to $200\text{m}$ keeping the same banking angle, the maximum safe speed on the road is

• A. $2v\text{m/s}$
• B. $0.5v\text{m/s}$
• C. $3v\text{m/s}$
• D. $4v\text{m/s}$

Answer 1

The correct option is A $2v\text{m/s}$


By the free body diagram and writing the force equation we get,

$N\cos \theta =mg...\left(i\right)$
$N\sin \theta =\frac{m{v}^2}{r}...\left(ii\right)$
Now by dividing both the above equations, we get,

$\theta ={\tan }^{-1}\left(\frac{v^2}{rg}\right)$
Condition for banked road is,
$\tan \theta =\frac{v^2}{rg}$, therefore, when the radius of curvature of the road is changed to $200\text{m}$ and $\theta$ remains the same we have,
$\frac{\frac{(v^{{}^{\prime }}{)}^2}{200\times g}}{\frac{v^2}{50\times g}}=1\Rightarrow (v^{{}^{\prime }}{)}^2=4v^2$
$\therefore$ The maximum safe speed on the road is
$v^{{}^{\prime }}=2vm/s$