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![### Banked Curves Dynamics
Two banked curves have the same radius. Curve A is banked at 13.9°, and curve B is banked at an angle of 18.1°. A car can travel around curve A without relying on friction at a speed of 13.7 m/s. At what speed can this car travel around curve B without relying on friction?
#### Explanation:
This problem involves the concept of banked curves, which are designed in such a way that vehicles can negotiate the curve at a particular speed without the need for frictional forces. This speed is often referred to as the "design speed". The system uses the banking angle and radius to calculate the ideal speed for the vehicle.
1. **Banking Angle (θ):** The tilt angle of the road or track curve.
2. **Radius of Curve (r):** The constant radius for both Curve A and Curve B.
3. **Speed (v):** The velocity at which the car travels around the curve relying solely on centripetal force without friction.
Given data:
- Curve A banking angle (θ₁) = 13.9°
- Curve B banking angle (θ₂) = 18.1°
- Speed on Curve A (v₁) = 13.7 m/s
- Radius (r) = constant
#### Formula:
The speed \( v \) of a vehicle on a banked curve without relying on friction is given by:
\[ v = \sqrt{r \cdot g \cdot \tan(\theta)} \]
So for Curve A:
\[ v_1 = \sqrt{r \cdot g \cdot \tan(\theta_1)} \]
We can isolate \( r \):
\[ r = \frac{v_1^2}{g \cdot \tan(\theta_1)} \]
For Curve B, we use the same formula:
\[ v_2 = \sqrt{r \cdot g \cdot \tan(\theta_2)} \]
Substitute the value of \( r \) from the equation of Curve A:
\[ v_2 = \sqrt{\left(\frac{v_1^2}{g \cdot \tan(\theta_1)}\right) \cdot g \cdot \tan(\theta_2)} \]
\[ v_2 = v_1 \cdot \sqrt{\frac{\tan(\](https://content.bartleby.com/qna-images/question/445dc1ba-0059-4453-a67d-7747e0259e94/c6ccdc94-4883-42d4-9b80-9e4bbf4e89b8/mzbrp4s_processed.png)
Transcribed Image Text:### Banked Curves Dynamics
Two banked curves have the same radius. Curve A is banked at 13.9°, and curve B is banked at an angle of 18.1°. A car can travel around curve A without relying on friction at a speed of 13.7 m/s. At what speed can this car travel around curve B without relying on friction?
#### Explanation:
This problem involves the concept of banked curves, which are designed in such a way that vehicles can negotiate the curve at a particular speed without the need for frictional forces. This speed is often referred to as the "design speed". The system uses the banking angle and radius to calculate the ideal speed for the vehicle.
1. **Banking Angle (θ):** The tilt angle of the road or track curve.
2. **Radius of Curve (r):** The constant radius for both Curve A and Curve B.
3. **Speed (v):** The velocity at which the car travels around the curve relying solely on centripetal force without friction.
Given data:
- Curve A banking angle (θ₁) = 13.9°
- Curve B banking angle (θ₂) = 18.1°
- Speed on Curve A (v₁) = 13.7 m/s
- Radius (r) = constant
#### Formula:
The speed \( v \) of a vehicle on a banked curve without relying on friction is given by:
\[ v = \sqrt{r \cdot g \cdot \tan(\theta)} \]
So for Curve A:
\[ v_1 = \sqrt{r \cdot g \cdot \tan(\theta_1)} \]
We can isolate \( r \):
\[ r = \frac{v_1^2}{g \cdot \tan(\theta_1)} \]
For Curve B, we use the same formula:
\[ v_2 = \sqrt{r \cdot g \cdot \tan(\theta_2)} \]
Substitute the value of \( r \) from the equation of Curve A:
\[ v_2 = \sqrt{\left(\frac{v_1^2}{g \cdot \tan(\theta_1)}\right) \cdot g \cdot \tan(\theta_2)} \]
\[ v_2 = v_1 \cdot \sqrt{\frac{\tan(\
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