Problem 9 - College Physics

Question

The Indy 500 . The Indianapolis Speedway (home of the Indy 500 ) consists of a 2.5 mile track having four turns, each 0.25 mile long and banked at \(9^{\circ} 12^{\prime} .\) What is the no-friction-needed speed (in \(\mathrm{m} / \mathrm{s}\) and mph) for these turns? (Do you think drivers actually take the turns at that speed?

Step-by-Step Solution

Answer

The no-friction speed is 18.64 m/s or 41.70 mph. Drivers usually take turns faster, using friction.

Step 1: Understanding the Problem

To find the no-friction-needed speed for a car making a turn on a banked track, we use the physics of circular motion. A turn on the track is 0.25 miles long, and the bank angle is given as \(9^\circ 12'\). We are required to calculate the speed in meters per second (m/s) and miles per hour (mph).

Step 2: Convert Angle to Decimal

First, convert the angle from degrees and minutes to a decimal. The angle is \(9^\circ 12'\). There are 60 minutes in a degree, so \(12' = \frac{12}{60} = 0.2\). Therefore, the angle in decimal degrees is \(9.2^\circ\).

Step 3: Use of Banked Curve Formula

The formula for the no-friction-required speed \(v\) on a banked curve is given by:\[ v = \sqrt{r \cdot g \cdot \tan(\theta)} \]where:- \(r\) is the radius of the turn,- \(g = 9.81 \text{ m/s}^2\) is the acceleration due to gravity,- \(\theta \) is the banking angle in radians. To convert \(\theta = 9.2^\circ\) to radians, use \(\theta = 9.2 \times \frac{\pi}{180}\).

Step 4: Calculate Radius and Convert It

Given each turn is 0.25 miles long (0.25 miles to meters is calculated as \(0.25 \times 1609.34 = 402.34 \text{ meters}\)), and assuming the turns form a perfect quarter-circle, the radius \(r\) of the turn can be found using the arc length formula: \(r = \frac{\text{arc length}}{\pi/2} = \frac{402.34}{1.57} \approx 256.30 \text{ meters}\).

Step 5: Calculate the No-Friction Speed (m/s)

Use the radius and converted angle from previous steps in the banked curve formula:\[ v = \sqrt{256.30 \cdot 9.81 \cdot \tan(9.2 \times \pi/180)} \]Calculate to get \(v\approx 18.64 \text{ m/s}\).

Step 6: Convert Speed to mph

Convert the speed from meters per second to miles per hour using the conversion:\(1 \text{ m/s} = 2.23694 \text{ mph}\).So, \(v = 18.64 \times 2.23694 \approx 41.70 \text{ mph}\).

Step 7: Interpret the Result

The no-friction-needed speed for the turn is approximately 18.64 m/s or 41.70 mph. In reality, drivers often take turns at higher speeds, relying on tire friction and skilled driving.

Key Concepts

Understanding a Banked Curve

A banked curve is a type of road or track design where the surface is tilted inwards towards the center of the curve. This inward tilt helps vehicles negotiate the curve more safely and efficiently by reducing reliance on the friction between the tires and the road surface. Instead, the normal force from the road itself helps provide some of the necessary centripetal force to keep a vehicle on the circular path.

When a vehicle travels through a banked curve, it experiences a combination of forces. The main forces at play are:

• The gravitational force, acting downwards.
• The normal force, which acts perpendicular to the surface of the banked track.
• The required centripetal force, which is necessary for the vehicle to follow the curved path.

These forces interact in such a way that, ideally, a car can take the turn at a particular speed without needing any additional frictional force from the tires. However, in reality, friction still plays an important role as drivers often choose to drive faster than the no-friction speed.

Angle Conversion from Degrees and Minutes

When dealing with banked curves, the inclination angle is often provided in degrees and minutes. This is a common way to measure angles, but in many mathematical contexts, it's easier to work with decimal degrees or radians, especially when plugging values into formulas like the one for banked curves.

To convert an angle given in degrees and minutes to decimal degrees:

• First, note that 1 degree is equal to 60 minutes.
• Convert the minutes to a fraction of a degree. For example, for the angle of \(9^{\circ} 12'\), divide 12 by 60 to get 0.2.
• Add this decimal to the degrees: \(9 + 0.2 = 9.2^{\circ}\).

Additionally, many equations in physics require angles to be in radians. To convert degrees to radians, multiply by \(\frac{\pi}{180}\). So, \(9.2^{\circ}\) becomes \(9.2 \times \frac{\pi}{180}\) in radians.

Calculating the Speed on a Banked Curve

The speed at which a car can travel around a banked curve without needing friction depends on several factors. These include the curve's radius, the banking angle, and gravitational force. The formula to determine this no-friction-required speed, \(v\), is given by:

• \(v = \sqrt{r \cdot g \cdot \tan(\theta)}\)
• Where \(r\) is the radius of the curve, \(g\) is the acceleration due to gravity (9.81 m/s²), and \(\theta\) is the banking angle in radians.

Once the values for the radius and the converted angle in radians are known, you can substitute them into the formula to find the speed. In our example, the speed was calculated to be approximately 18.64 m/s.

Understanding this formula is crucial for optimizing the design of roadways, especially high-speed tracks, where maintaining a safe and efficient cornering speed is essential.

Radius Calculation Using Arc Length

The radius of a curve is a vital component for calculating the speed on a banked curve. When you know the arc length of a turn and assume that the turn is part of a perfect circle, you can calculate the radius using the formula:

• \(r = \frac{\text{arc length}}{\theta}\), where \(\theta\) is in radians.

In the example of the Indy 500 track, each turn has an arc length of 0.25 miles. First, convert the arc length to meters, resulting in approximately 402.34 meters. Since the turn is one-fourth of a full circle, \(\theta\) for a quarter-turn is \(\frac{\pi}{2}\) radians.

The radius formula simplifies to \(r = \frac{402.34}{1.57}\) meters, resulting in an approximate radius of 256.30 meters. This calculation is important as it underscores the geometric relationship between the curve's arc length and its radius, both of which influence the ability to navigate the curve at specific speeds.