Running in a wind A strong west wind blows across a circular...

Running in a wind 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. (FIGURE CAN'T COPY) 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?

Running in a wind 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. (FIGURE CAN'T COPY)
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?

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Key Concepts

Trigonometric Variation of Speed

This concept involves representing variable speeds using trigonometric functions such as cosine. It models how external factors (like wind) cyclically influence an object's speed as it moves along a circular path, causing acceleration when moving with the wind and deceleration when moving against it.

Average Value of a Function

In calculus, the average value of a function over an interval is found by integrating the function over that interval and then dividing by the interval's length. This technique is crucial when dealing with variable speeds to determine an overall average speed across a complete cycle.

Circular Motion and Angular Displacement

In the context of circular motion, a runner's position can be parameterized by a central or angular displacement. The relationship between the angular displacement and arc length (using the radius of the circle) is fundamental for converting between angular measurements and actual distances covered on the track.

Relative Motion in an External Medium

This concept addresses how the speed of an object is affected by an external medium—in this case, wind. The effective speed is the result of the runner’s own effort combined with the additive or subtractive effect of wind, depending on the direction of motion relative to the wind.

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