Two banked curves on a highway have the same radius. Curve A is banked at 13 degrees and highway B is banked at 19 degrees. A car can travel around curve A without relying on friction at a speed of 18 m/s. At what speed can this car travel around curve B without relying on friction?

Question

Two banked curves on a highway have the same radius. Curve A is banked at 13 degrees and highway B is banked at 19 degrees. A car can travel around curve A without relying on friction at a speed of 18 m/s. At what speed can this car travel around curve B without relying on friction?

Answer 1

A roadway is designed for traffic moving at a speed of 68 m/s. A curved section of the roadway is a circular arc of 140 m radius. The roadway is banked--so that a vehicle can go around the curve--with the lateral friction forces equal to zero. The angle at which the roadway is banked is closest to:

Select one:
a. 73°
b. 67°
c. 75°
d. 69°
e. 71°

Answer 2

To determine the speed at which a car can travel around curve B without relying on friction, we need to apply the concept of centripetal force and the equation for frictionless banking of curves.

The equation for frictionless banking is given by:

tan(θ) = v^2 / (g * r)

Where:
- θ is the angle of the banked curve,
- v is the velocity of the car,
- g is the acceleration due to gravity (approximately 9.8 m/s^2),
- r is the radius of the banked curve.

We are given that curve A is banked at 13 degrees and the car can travel around it at a speed of 18 m/s. We need to find the speed the car can travel around curve B, which is banked at 19 degrees.

Let's assume the radius of both curves is the same.

For curve A:
θA = 13 degrees
vA = 18 m/s

Using the equation for curve A, we can rearrange it to solve for rA:
rA = vA^2 / (g * tan(θA))

Next, we can use the same equation for curve B, where θB = 19 degrees, to solve for vB:
vB = sqrt(rB * g * tan(θB))

To find vB, we need to determine the radius of curve B. Since both curves have the same radius, we can equate the expressions for the radii of curve A and curve B:

rA = rB

Substituting the expression for rA into the equation for curve B, we have:

sqrt(rA * g * tan(θB)) = vB

We can now substitute the known values into the equation and solve for vB.

1. Calculate rA:
rA = (18 m/s)^2 / (9.8 m/s^2 * tan(13 degrees))

2. Calculate rB:
rB = rA

3. Calculate vB:
vB = sqrt(rB * 9.8 m/s^2 * tan(19 degrees))

By following these calculations, we can determine the speed at which the car can travel around curve B without relying on friction.

Two banked curves on a highway have the same radius. Curve A is banked at 13 degrees and highway B is banked at 19 degrees. A car can travel around curve A without relying on friction at a speed of 18 m/s. At what speed can this car travel around curve B without relying on friction?

A roadway is designed for traffic moving at a speed of 68 m/s. A curved section of the roadway is a circular arc of 140 m radius. The roadway is banked--so that a vehicle can go around the curve--with the lateral friction forces equal to zero. The angle at which the roadway is banked is closest to:

14.7 m/s

To determine the speed at which a car can travel around curve B without relying on friction, we need to apply the concept of centripetal force and the equation for frictionless banking of curves.