Question

A race car undergoes uniform circular motion around a banked race track with a radius of 245 m. The track is banked at an angle of θ = 26.0°towards the centre of the track. The coefficient of static friction between the track and the tires is 0.880. What is the maximum speed at which a car can still safely make the curve?

Answer 1

The maximum speed at which a car can safely make the curve is approximately 44.6 m/s.

Step-by-step explanation:

Identify the relevant forces:

There are three main forces acting on the car:

Gravitational force (mg) downwards

Normal force (N) perpendicular to the track surface

Centripetal force (Fc) directed inwards towards the center of the curve

Resolve the normal force:

The normal force can be resolved into two components:

Vertical component (Nv) equal to mg cosθ

Horizontal component (Nh) equal to mg sinθ

Apply Newton's second law for the horizontal direction:

Nh = Fc

mg sinθ = mv²/r

v² = (g sinθ) * r

Apply Newton's second law for the vertical direction:

Nv + F_friction = mg

mg cosθ + µ_s * Nh = mg

F_friction = µ_s * Nh = µ_s * mg sinθ

5. Substitute F_friction into the horizontal force equation:

v² = (g sinθ) * r * (1 - µ_s cosθ)

Substitute the given values and calculate the maximum speed:

g ≈ 9.81 m/s²

θ = 26.0°

r = 245 m

µ_s = 0.880

v² = (9.81 m/s² * sin 26.0°) * 245 m * (1 - 0.880 * cos 26.0°)

v ≈ 44.6 m/s

Therefore, the maximum speed at which a car can safely make the curve is approximately 44.6 m/s.