Question

What is the safety speed for a vehicle moving along a curved horizontal banked road?

Hint:

The safety speed on a banked curve depends only on the radius of the curve, the acceleration due to gravity, and the banking angle. Friction is not considered in this case.

Answer 1

Step 1: Forces acting on the vehicle.
For a vehicle moving on a banked curve, the forces acting are: Gravitational force (\( mg \)). Normal reaction force (\( N \)). Frictional force (\( f \)). The net force provides the necessary centripetal force for circular motion. Step 2: Resolving forces.
The component of the normal reaction along the radius provides the centripetal force: \[ N \sin \theta = \frac{m v^2}{r}, \] where \( v \) is the speed, \( r \) is the radius of the curve, and \( \theta \) is the banking angle. Step 3: Frictionless case (Safety speed).
For the safety speed, assume friction is negligible. The normal force's component balances the centripetal force: \[ v^2 = r g \tan \theta. \] Solve for \( v \): \[ v = \sqrt{r g \tan \theta}. \] Step 4: Final Answer.
The safety speed is: \[ \boxed{V = \sqrt{r g \tan \theta}}. \]