Two banked curves have the same radius. Curve A is banked at 11.0 °, and curve B is banked at an angle of 17.4 °. A car can travel around curve A without relying on friction at a speed of 15.3 m/s. At what speed can this car travel around curve B without relying on friction?

r of A = r of B

θ of A = 11°
θ of B = 17.4°
v of A = 15.3 m/s
v of B = ?

tan(θA) = (vA^2)/(rg)
derive to: r = (vA^2)/(tan(θA))(g)
r=(15.3^2)/(tan11)(9.8)
r=122.8866 m

tan(θB) = (vB^2)/(rg)
derive to: vB=sqrt[(r)(tanθB)(g)]
vB=sqrt[(122.8866)(tan17.4)(9.8)]
vB=19.4268 m/s

Answer: vB = 19.43 m/s

To find the speed at which the car can travel around curve B without relying on friction, we need to use the concept of centripetal force and the component of the car's weight that provides this force.

1. Start by calculating the centripetal force on the car traveling around curve A. The centripetal force (Fc) is given by the equation:

Fc = m * v^2 / r

Where:
- m is the mass of the car
- v is the speed of the car
- r is the radius of the curve

2. Since the radius of curve A and curve B is the same, we can use the same equation to find the centripetal force on curve B as well.

3. The component of the car's weight that provides the centripetal force is given by the equation:

Fc = m * g * sin(θ)

Where:
- g is the acceleration due to gravity
- θ is the angle of the banked curve

4. Equate the two expressions for centripetal force:

m * v_A^2 / r = m * g * sin(θ_B)

Where:
- v_A is the speed of the car on curve A
- θ_B is the angle of the banked curve B

5. Rearrange the equation to solve for the speed of the car on curve B:

v_B = sqrt(r * g * sin(θ_B) / sin(θ_A))

Plug in the given values:
- θ_A = 11.0°
- θ_B = 17.4°
- v_A = 15.3 m/s

6. Calculate the speed of the car on curve B:

v_B = sqrt(r * g * sin(17.4°) / sin(11.0°))

Note: Make sure to convert the angles from degrees to radians when using a scientific calculator or programming language that expects angles in radians.

The value of v_B will be the speed at which the car can travel around curve B without relying on friction.