Final answer:
The speed at which a car can travel around curve B without relying on friction can be determined using the concept of an ideally banked curve, by setting up a ratio between the tangents of the banking angles of curve A and curve B and the speeds for each curve. Given the information for curve A, we can solve for the ideal speed for curve B.
Step-by-step explanation:
To determine the speed at which a car can travel around curve B without relying on friction, we can use the concept of an ideally banked curve. An ideally banked curve allows a vehicle to travel through the curve at a certain speed without friction by carefully balancing the gravitational force and the centripetal force required to make the turn. For curve A, we are given that the car travels at 18 m/s at a banking angle of 13°. Now, we can apply the following formula derived from the centripetal force:
\( v = \sqrt{r \cdot g \cdot tan(\theta)} \)
Where:
- v is the ideal speed
- r is the radius of the curve
- g is the acceleration due to gravity (9.8 m/s²)
- \(\theta\) is the bank angle
Since the radius (r) and gravitational force (g) are constant for curves A and B, we can set up a ratio between the speeds and the tangents of the banking angles:
\( \frac{v_A}{v_B} = \sqrt{\frac{tan(\theta_A)}{tan(\theta_B)}} \)
Plug in the known values:
\( \frac{18}{v_B} = \sqrt{\frac{tan(13^\circ)}{tan(19^\circ)}} \)
Solving for v_B gives us the ideal speed for curve B.