Question

A strong west wind blows across a circular running track. Abe and Bess start at the south end of the track and at the same time, Abe starts running clockwise and Bess starts running counterclockwise. Abe runs with a speed (in units of miles/hour) given by \(u(\varphi)=3-2 \cos \varphi\) and Bess runs with a speed given by \(v(\theta)=3+2 \cos \theta,\) where \(\varphi\) and \(\theta\) are the central angles of the runners. a. Graph the speed functions \(u\) and \(v,\) and explain why they describe the runners' speeds (in light of the wind). b. Compute each runner's average speed (over one lap) with respect to the central angle. c. Challenge: If the track has a radius of \(\frac{1}{10} \mathrm{mi}\), how long does it take each runner to complete one lap and who wins the race?

Short answer

The average speed of Abe and Bess with respect to the central angle is 3 miles/hour each. To complete one lap, both Abe and Bess will take the same time, which is (2π * (1/10) mile)/3. As both runners have the same average speed and take the same time to complete one lap, there is no winner, and they will reach the starting point simultaneously.

Step-by-step solution

Part b: Compute the average speed of each runner with respect to the central angle.

To calculate the average speed, we need to integrate each runner's speed function over the interval of the central angle [0, 2π] and divide the result by 2π. 1. Integrate Abe's speed function \(u(\varphi)\): $$ \bar{u} = \frac{1}{2\pi}\int_{0}^{2\pi} (3 - 2 \cos \varphi) d\varphi $$ 2. Integrate Bess' speed function \(v(\theta)\): $$ \bar{v} = \frac{1}{2\pi}\int_{0}^{2\pi} (3 + 2 \cos \theta) d\theta $$ Now, we will compute the integrals. 3. Compute the integral for Abe's average speed: $$ \bar{u} = \frac{1}{2\pi} \left[ 3\varphi - 2 \sin \varphi \right]_{0}^{2\pi} $$ 4. Compute the integral for Bess' average speed: $$ \bar{v} = \frac{1}{2\pi} \left[ 3\theta + 2 \sin \theta \right]_{0}^{2\pi} $$ 5. Evaluate the expressions at the limits: $$ \bar{u} = \frac{1}{2\pi} (3(2\pi) - 0) = 3 $$ $$ \bar{v} = \frac{1}{2\pi} (3(2\pi) - 0) = 3 $$ Abe and Bess have the same average speed of 3 miles/hour with respect to the central angle.

Part c: Calculate the time to complete one lap, and determine the winner.

To find the time each runner takes to complete one lap, we will use their respective speed functions and the track's radius. 1. Find the circumference of the track: $$ C = 2\pi (\frac{1}{10} \text{mi}) $$ 2. Find the time it takes Abe to complete one lap: $$ t_A = \frac{C}{\bar{u}} = \frac{2\pi (\frac{1}{10} \text{mi})}{3} $$ 3. Find the time it takes Bess to complete one lap: $$ t_B = \frac{C}{\bar{v}} = \frac{2\pi (\frac{1}{10} \text{mi})}{3} $$ 4. Determine the winner: Since Abe and Bess have the same average speed and the time it takes for them to complete one lap is the same, there is no winner in this race. Abe and Bess will reach the starting point simultaneously.

Key concepts

Integration

Integration is a fundamental concept in calculus that is used extensively to solve problems involving accumulation. In this exercise, we need to find the average speed of the runners, which requires calculating the integral of their speed functions over a complete lap of the track. Here's the process broken down:

• The speed functions for Abe and Bess are given by \(u(\varphi)=3-2 \cos \varphi\) and \(v(\theta)=3+2 \cos \theta\), respectively.
• To find their average speeds, we integrate these functions over the angle from 0 to \(2\pi\), the full circular track.

This integral gives the total miles traveled with respect to the central angle. Since we want the average speed, we divide this accumulated value by the full angle \(2\pi\). Here's how it works:

• For Abe: \(\bar{u} = \frac{1}{2\pi}\int_{0}^{2\pi} (3 - 2 \cos \varphi) d\varphi\).
• For Bess: \(\bar{v} = \frac{1}{2\pi}\int_{0}^{2\pi} (3 + 2 \cos \theta) d\theta\).

Upon evaluating these integrals, both average speeds turn out to be 3 mph, indicating that over a full lap, Abe and Bess both maintain the same average speed.

Circular Motion

Circular motion involves the movement of an object along the circumference of a circle or a circular path. In this exercise, both Abe and Bess are running on a circular track. Their motion is influenced by how the wind affects their speeds at different points around the circle.

• Abe runs clockwise, facing more resistance when moving directly against the wind, which is factored into his speed function \(u(\varphi)\).
• Bess, moving counterclockwise, runs with an aided speed when the wind is behind her, influencing her speed function \(v(\theta)\).

These speed functions alter based on the cosine component, either increasing or decreasing their speed depending on their position on the track as dictated by their central angle. Such alterations are common in circular motion problems where external forces like wind can impact speed irregularly around the path.
Understanding how these factors affect their journey gives insight into realistic modeling of movement in a circular track scenario.

Graphing Functions

Graphing functions is a crucial method in visualizing mathematical relationships, particularly in understanding how speed changes over time or distance. In their running challenge, viewing the graphs of \(u(\varphi)\) and \(v(\theta)\) can visually reinforce how the wind affects Abe's and Bess's speeds.

• The function \(u(\varphi) = 3 - 2 \cos \varphi\) represents Abe's speed. It will show periodic variations, typically dipping where the wind resistance is maximum.
• The function \(v(\theta) = 3 + 2 \cos \theta\) for Bess indicates where the wind boosts her speed, peaking similarly according to her direction relative to the wind.

These graphs offer tangible insight, allowing us to anticipate and interpret changes in speed at various points along their run. This exercise underscores how equations can manifest in real-world scenarios, and how graphing helps elucidate complex dynamics like those of wind-influenced running speeds. Using graphs not only assists in solving the problem but also deepens understanding of the concepts at play.