A car enters a banked curve with a radius of 100 meters. If the banking angle is 45 degrees, calculate the car's speed at which it can navigate the curve without relying on friction.
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To determine the speed at which the car can navigate the curve without relying on friction, we need to consider the gravitational force and the centripetal force acting on the car.
The gravitational force acts vertically downwards and can be split into two components: the force acting perpendicular to the plane of the curve (mgcosθ), and the force acting parallel to the plane of the curve (mgsinθ), where m is the mass of the car and θ is the banking angle.
The centripetal force acts towards the center of the curve and is provided solely by the horizontal component of the gravitational force (mgsinθ).
Since the gravitational force provides the centripetal force in this case (no friction), we can set the two forces equal to each other:
mgsinθ = mv^2/r
Where v is the speed of the car and r is the radius of the curve.
Rearranging the equation, we can solve for v:
v^2 = rgsinθ
v = √(rgsinθ)
Plugging in the given values, we have:
v = √(100 * 9.8 * sin(45°))
v = √(980 * 0.7071)
v = √(694.498)
v ≈ 26.34 m/s
Therefore, the car's speed at which it can navigate the curve without relying on friction is approximately 26.34 m/s.
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