This problem involves a car driving on a banked curve with no friction, which is a common type of physics problem related to circular motion.

Given:

• Angle of the bank, $\theta = 30^\circ$
• Radius of the curve, $R = 50.0 \, \text{m}$

To find:

• The fastest speed $v$ at which the car can drive without friction.

Formula:

For a frictionless banked curve, the fastest speed the car can travel is determined by balancing the gravitational force and the centripetal force. The formula for the maximum speed $v$ is:

$v = \sqrt{R \cdot g \cdot \tan(\theta)}$

Where:

• $R = 50.0 \, \text{m}$ is the radius,
• $g = 9.8 \, \text{m/s}^2$ is the acceleration due to gravity,
• $\theta = 30^\circ$ is the bank angle.

Solution:

1. Calculate $\tan(30^\circ)$:

$\tan(30^\circ) = \frac{1}{\sqrt{3}} \approx 0.577$

2. Plug in the values into the formula:

$v = \sqrt{50.0 \, \text{m} \times 9.8 \, \text{m/s}^2 \times 0.577}$

$v = \sqrt{283.73} \approx 16.84 \, \text{m/s}$

Conclusion:

The fastest speed at which the car can drive is approximately 16.8 m/s, which matches option (d).

Let me know if you'd like more details on this or have any questions! Here are 5 related questions for further exploration:

1. How would the speed change if the radius of the curve were doubled?
2. What would happen if friction were introduced on the road surface?
3. How would the speed change if the angle of the bank were increased to 45 degrees?
4. Can a car travel faster than the calculated speed on this banked curve? Why or why not?
5. What role does centripetal force play in a banked curve problem?

Tip: Remember that on banked curves, higher angles allow for higher speeds without relying on friction.