Question

A curve has a radius of $50\text{m}$ and a banking angle ($\theta$) of ${15}^{\circ }$. What is the maximum safe speed for a car on this curve if the road does not offer any friction? Take $g=10{\text{m/s}}^2$.

• A. $10\sqrt{5(4-\sqrt{3})}\text{m/s}$
• B. $10\sqrt{5(2-\sqrt{3})}\text{m/s}$
• C. $10\sqrt{10(2-\sqrt{5})}\text{m/s}$
• D. $10\sqrt{5(3-\sqrt{3})}\text{m/s}$

Answer 1

The correct option is B $10\sqrt{5(2-\sqrt{3})}\text{m/s}$

Applying the condition of banking for a frictionless road:
$\tan \theta =\frac{v^2}{rg}$

$\Rightarrow v^2=rg\tan \theta$
$=50\times 10\times \tan {15}^{\circ }$
$=50\times 10\times \tan ({45}^{\circ }-{30}^{\circ })$
$=50\times 10\times \frac{\tan {45}^{\circ }-\tan {30}^{\circ }}{1+\tan {45}^{\circ }\times \tan {30}^{\circ }}$
$=50\times 10\times \frac{\sqrt{3}-1}{\sqrt{3}+1}$

Multiply by $\sqrt{3}-1$ on both numerator and denominator
$v^2=500(2-\sqrt{3})$
$\therefore v=10\sqrt{5(2-\sqrt{3})}\text{m/s}$