Question

A curve has a radius of 50 meters and a banking angle of 15. What is the ideal, or critical, speed (the speed for which no friction is required between the car’s tires and the surface) for a car on this curve?
\[A.\,20{m}/{s}\;\]
\[B.\,5{m}/{s}\;\]
\[C.\,11{m}/{s}\;\]
\[D.\,25{m}/{s}\;\]

Answer 1

Hint: This is a direct question. The formula that relates the critical speed of the car, the radius of the curve and the banking angle should be used to solve this problem. As we are given with the values, upon substituting these values in the formula we will obtain the value of the critical speed of the car.

Formula used: \[v=\sqrt{rg\tan \theta }\]

Complete step by step answer:
Let us first understand the parameters given.
The radius of the curve is 50 meters, the curve refers to the circular road along which the car moves.
A banking angle of 15. The banking angle refers to the inclination of the horizontal component of the road with respect to the ground. This banking angle provides the centrifugal force to the vehicles to keep them in motion along a circular path.
The critical speed in the absence of the friction for a vehicle without overturning is given by a formula as follows.
\[v=\sqrt{rg\tan \theta }\]
Where r is the radius of the curve, g is the gravitational constant and \[\theta \]is the angle made by the horizontal component of the road with the ground (that is, the banking angle).
Substitute the given values in the above equation to find the value of the critical or the ideal speed. So, we get,
\[\begin{align}
  & v=\sqrt{50\times 9.8\times \tan 15{}^\circ } \\
 & \Rightarrow v=\sqrt{131.29} \\
 & \Rightarrow v=11.45\,{m}/{s}\; \\
\end{align}\]
This value of the critical speed can be rounded off to the nearest decimal as follows.
\[v=11\,{m}/{s}\;\]

So, the correct answer is “Option C”.

Note: The units of the parameters should be taken care of. Even with other parameters such as the banking angle, the radius of the curve can be asked by providing the values of the other parameters. In this problem, the curve refers to the circular path/road.