A newly completed highway has a curve with a radius of 130 m and is banked at an angle of 24.8 degrees. What is the maximum speed that a minivan can have and still follow the curve safely under very icy conditions?

24 m/s

Explanation:

The maximum speed that a minivan can have and still follow the curve safely under very icy conditions is 24 m/s.

g = 9.8 m/s², r = 130 m, and θ = 24.8°:

v = √ (9.8 m/s² * 130 m * tan 24.8°)

v = 24.3 m/s

24 m/s

Explanation:

Let's say that under very icy conditions, there is no friction.

Draw a free body diagram. There are 2 forces acting on the van. Gravity straight down, and normal force perpendicular to the surface.

Sum of the forces in the radial (+x) direction:

∑F = ma

N sin θ = m v² / r

Sum of the forces in the y direction:

∑F = ma

N cos θ - mg = 0

Solve for N in the second equation:

N cos θ = mg

N = mg / cos θ

Substitute into the first equation:

(mg / cos θ) sin θ = m v² / r

mg tan θ = m v² / r

g tan θ = v² / r

v = √ (gr tan θ)

Given g = 9.8 m/s², r = 130 m, and θ = 24.8°:

v = √ (9.8 m/s² * 130 m * tan 24.8°)

v = 24.3 m/s

Rounded to two significant figures, the maximum velocity is 24 m/s (approximately 54 mph).