If a car takes a banked curve at less than the ideal speed, friction is needed to keep it from sliding toward the inside of the curve (a real problem on icy mountain roads).

Question

If a car takes a banked curve at less than the ideal speed, friction is needed to keep it from sliding toward the inside of the curve (a real problem on icy mountain roads).

(a) Calculate the ideal speed to take a 105-m radius curve banked at 13.0°.

km/h

(b) What is the minimum coefficient of friction needed for a frightened driver to take the same curve at 20.0 km/h?

Answer 1

(a) To calculate the ideal speed, we can make use of the formula for the ideal speed of a car on a banked curve:

v = √(g * r * tan(θ))

where:
v = ideal speed
g = acceleration due to gravity (approximately 9.8 m/s²)
r = radius of the curve
θ = angle of the bank

First, we need to convert the radius from meters to kilometers:
105 m = 0.105 km

Next, we can substitute the given values into the formula:

v = √(9.8 m/s² * 0.105 km * tan(13.0°))

To calculate this, we need to use the tangent function. However, most calculators work with radians, so we'll need to convert the angle from degrees to radians:

θ (in radians) = θ (in degrees) * π / 180

θ (in radians) = 13.0° * π / 180 = 0.2269 radians

Now, we can substitute in the values and solve for v:

v = √(9.8 m/s² * 0.105 km * tan(0.2269 radians))

Using a calculator, we find:

v ≈ 29.5 km/h

Therefore, the ideal speed to take the 105-m radius curve banked at 13.0° is approximately 29.5 km/h.

(b) To find the minimum coefficient of friction needed for a frightened driver to take the same curve at 20.0 km/h, we can make use of the following equation:

μ = (v² - v₀²)/(g * r * tan(θ))

where:
μ = coefficient of friction
v = speed of the car
v₀ = ideal speed
g = acceleration due to gravity
r = radius of the curve
θ = angle of the bank

First, we convert the speed from km/h to m/s:

20.0 km/h = 20.0 * (1000 m / 1 km) / (3600 s / 1 h) ≈ 5.56 m/s

Next, we substitute the given values into the equation:

μ = (5.56² km/s - 29.5² km/s) / (9.8 m/s² * 0.105 km * tan(13.0°))

Since the radius is already in kilometers, we don't need to convert it. By using the tangent function to calculate the angle in radians as we did in part (a), we can substitute the values and solve for μ:

μ ≈ (5.56² km/s - 29.5² km/s) / (9.8 m/s² * 0.105 km * tan(0.2269 radians))

Using a calculator, we obtain:

μ ≈ 0.088

Therefore, the minimum coefficient of friction needed for a frightened driver to take the same curve at 20.0 km/h is approximately 0.088.