To calculate the minimum speed the car can navigate the curve without relying on friction, we need to use the concept of the normal force and the force of gravity.

In a banked curve, the normal force is split into two components: a vertical component (Fn⊥) that opposes the force of gravity, and a horizontal component (Fn‖) that provides the necessary centripetal force for the car to stay on the curved path.

Given:
Radius of the curve (r) = 100 meters
Banking angle (θ) = 30 degrees
Acceleration due to gravity (g) = 9.8 m/s²

We can calculate the vertical component of the normal force (Fn⊥) using trigonometry. The vertical component counteracts the gravitational force (mg):

Fn⊥ = mg

We can express the gravitational force as:

mg = m * g

where m is the mass of the car and g is the acceleration due to gravity.

Next, we'll calculate the horizontal component of the normal force (Fn‖). The horizontal component provides the centripetal force needed for the car to navigate the curve:

Fn‖ = m * v² / r

where v is the speed of the car.

Since the car is not relying on friction, the horizontal component of the normal force (Fn‖) provides the necessary centripetal force:

Fn‖ = m * v² / r = m * ω² * r

where ω is the angular velocity.

From the conservation of energy, we know that the centripetal force (Fn‖) is equal to the gravitational force (mg):

m * v² / r = m * g

Simplifying and solving for v:

v² / r = g
v² = r * g
v = sqrt(r * g)

Plugging in the given values:

v = sqrt(100 * 9.8) ≈ sqrt(980) ≈ 31.30 m/s

Therefore, the minimum speed the car can navigate the curve without relying on friction is approximately 31.30 m/s.

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