Suppose the rated speed of a highway curve of 300-ft radius is 40 miles per hour. – [Free] B22
Suppose the rated speed of a highway curve of 300-ft radius is 40 miles per hour. If the coefficient of friction between the tires and the road is 0.60 , what is the maximum speed at which a car can round the curve without skidding?
![Suppose the rated speed of a highway curve of 300-ft radius is 40 miles per hour. - [Free] B22](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20482%20245'%3E%3C/svg%3E)
Answer
🛣️ Maximum Speed on a Banked Highway Curve Without Skidding
🔹 Given Data
- Radius of the curve: r = 300 ft
- Rated speed: vₛ = 40 mph
- Coefficient of friction: μ = 0.60
- Acceleration due to gravity: g = 32.19 ft/s²
- Conversion: 1 mile = 5280 ft, 1 hour = 3600 s
🟦 Step 1: Convert Rated Speed
vr = 40 × (5280 / 3600) = 58.67 ft/s
🟨 Step 2: Calculate Banking Angle (θ)
ac = v² / r
tan(θ) = v² / (rg)
tan(θ) = (58.67)² / (300 × 32.19) = 0.3564
🟩 Step 3: Use Friction to Find Maximum Speed
Equilibrium Equations:
Nsin(θ) + fcos(θ) = mv² / r
Ncos(θ) − fsin(θ) = mg
f = μN
v²max / (rg) = (sin(θ) + μcos(θ)) / (cos(θ) − μsin(θ))
v²max = rg × (tan(θ) + μ) / (1 − μtan(θ))
v²max = 300 × 32.19 × (0.3564 + 0.6) / (1 − 0.6 × 0.3564)
v²max = 9657 × (0.9564 / 0.78616) = 11748.187
🟥 Step 4: Compute Final Speed
vmax = √11748.187 = 108.389 ft/s
vmax = 108.389 × (3600 / 5280) = 73.90 mph
✅ Final Answer
The maximum speed at which a car can round the curve without skidding is:
73.90 miles per hour