On a banked race track, the smallest circular path on which cars can move has a radius of 122 m, while the largest has a radius of 164 m, as the drawing illustrates. The height of the outer wall is 17.1 m. Find (a) the smallest and (b) the largest speed at which cars can move on this track without relying on friction.

This problem is about the motion of cars on a banked curve, where we need to calculate the smallest and largest speeds at which cars can move without relying on friction. In this case, the forces involved are the gravitational force and the normal force provided by the banked surface.

### Given:
- Radius of the smallest circular path: \( R_{\text{min}} = 122 \, \text{m} \)
- Radius of the largest circular path: \( R_{\text{max}} = 164 \, \text{m} \)
- Height of the outer wall (banking height): \( h = 17.1 \, \text{m} \)

### Assumptions and Formula:
The speed of a car on a frictionless banked curve can be calculated using the formula:
\[
v = \sqrt{R \cdot g \cdot \tan(\theta)}
\]
Where:
- \( v \) is the speed of the car,
- \( R \) is the radius of the circular path (122 m for the smallest path and 164 m for the largest path),
- \( g = 9.8 \, \text{m/s}^2 \) is the acceleration due to gravity,
- \( \theta \) is the banking angle of the track.

#### To find the banking angle \( \theta \), we use the geometry of the banked track. The height \( h \) of the outer wall and the difference between the largest and smallest radii can be used to find \( \theta \):

\[
\tan(\theta) = \frac{h}{R_{\text{max}} - R_{\text{min}}}
\]

Now, let's calculate \( \theta \) and then find the smallest and largest speeds for the given radii.

### Step 1: Calculate \( \theta \)
\[
\tan(\theta) = \frac{17.1}{164 - 122}
\]
\[
\tan(\theta) = \frac{17.1}{42} \approx 0.4071
\]
\[
\theta = \tan^{-1}(0.4071) \approx 22.17^\circ
\]

### Step 2: Calculate the smallest and largest speeds

For the smallest radius (\( R_{\text{min}} = 122 \, \text{m} \)):
\[
v_{\text{min}} = \sqrt{122 \times 9.8 \times \tan(22.17^\circ)}
\]

For the largest radius (\( R_{\text{max}} = 164 \, \text{m} \)):
\[
v_{\text{max}} = \sqrt{164 \times 9.8 \times \tan(22.17^\circ)}
\]

Let me compute these values.The speeds at which cars can move on the banked track without relying on friction are as follows:

- (a) The smallest speed on the inner path: \( v_{\text{min}} \approx 22.06 \, \text{m/s} \)
- (b) The largest speed on the outer path: \( v_{\text{max}} \approx 25.58 \, \text{m/s} \)

Would you like further details or explanations?

Here are 5 related questions for further exploration:
1. How does increasing the banking angle affect the required speed for frictionless motion?
2. What is the effect of increasing the height of the outer wall on the speed limits?
3. How would the speeds change if the track were completely flat?
4. Can friction allow cars to exceed these speeds? How would you calculate it?
5. What happens to the forces on the car if the track is wet or icy?

**Tip:** The banking angle is crucial for safe, frictionless turning—steeper angles allow higher speeds!

This detailed guide explores how to calculate the smallest and largest speeds at which cars can move on a banked race track without relying on friction. Given the radii of 122 m and 164 m, and a height of 17.1 m, we use principles of circular motion and trigonometry to find these speeds. Suitable for students in Grades 11-12.

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