Question

Calculate the maximum speed with which a car can be driven safely along a curved road of radius 30 m and banked at 30 ° with the horizontal. Given, $g=9.8m{{s}^{-2}}$
$\begin{align}
  & \text{A}\text{. }13.03m{{s}^{-1}} \\
 & \text{B}\text{. }12.08m{{s}^{-1}} \\
 & \text{C}\text{. }15.08m{{s}^{-1}} \\
 & \text{D}\text{. }14.02m{{s}^{-1}} \\
\end{align}$

Answer 1

- Hint: We know that when the road is banked the horizontal component of the normal reaction provides the necessary centripetal force. Therefore we will put simply the value in the formula of maximum velocity on a banked road and we will obtain the answer.

Formula used:
 We know the formula for maximum velocity on banking road is given as;
\[v=\sqrt{rg\tan \theta }\]

Complete step-by-step solution
Given that radius $r=30m\text{ and }\theta ={{30}^{{}^\circ }}$
 We know that
The maximum speed on car can be driven safely along the curve is given by
 \[\begin{align}
  & v=\sqrt{rg\tan \theta }=\sqrt{30\times 9.8\times \tan ({{30}^{{}^\circ }})} \\
 & =\sqrt{30\times 9.8\times \dfrac{1}{\sqrt{3}}}=13.03m{{s}^{-1}} \\
\end{align}\]
The maximum speed with which a car can be driven safely along a curved road is \[13.03m{{s}^{-1}}\].
Hence the correct option is A.

Additional information:
Banking of roads is defined because the phenomenon during which the sides are raised for the curved roads above the inner edge to supply the required force to the vehicles so that they take a safe turn. Now, let us recall, what is centripetal force? It is the force that pulls or pushes an object toward the middle of a circle because it travels, causing angular or circular motion. In the next few sections, allow us to discuss the angle of banking and therefore the terminologies utilized in banking of roads. Other terminologies used are banked turn which is defined because the turn or change of direction during which the vehicle inclines towards inside. The angle at which the vehicle is inclined is defined because of the bank angle. The inclination happens at the longitudinal and horizontal axis.

 Note: The large amount of friction between the tyres and the road produces considerable wear and tear of the tyres. And to avoid this situation the curved roads are given inclination sloping upwards and outwards to the circumference. This reduces the tearing of the tyres because the horizontal component of the normal reaction provides the necessary centripetal force. The system of raising the outer edge of a curved road above its inner edge is called banking of road and angle of banking is defined as the angle through which the outer edge of the curved road is raised above the inner edge is called angle of banking.