Final answer:
The tracks banked at a 5.0° angle with a radius of 500.0 m are designed for trains moving at an ideal speed of 29.4 m/s, which is computed using the formula for centripetal force on a frictionless, banked curve.
Step-by-step explanation:
The question pertains to the physical concept of centripetal force and the ideal speed for a train on banked tracks. To determine the ideal speed for a train on tracks banked at a 5.0° angle with a radius of 500.0 m, we use the relationship between the bank angle (θ), the radius of the curve (r), and the gravitational acceleration (g). On a banked curve with no friction, the ideal speed (v) can be calculated using the formula v² = r*g*tan(θ).
Plugging in the values r = 500.0 m, g = 9.8 m/s², and θ = 5.0° into the formula, we first convert θ to radians and then solve for v:
v² = 500.0 m * 9.8 m/s² * tan(5.0°)
v² = 4900 m²/s² * tan(5.0°)
v = √(4900 m²/s² * tan(5.0°))
v ≈ 29.4 m/s
Therefore, the tracks are designed for trains moving at a speed of 29.4 m/s.