A race Jean and Juan run a one-lap race on a circular track. Their angular positions on the track during the race are given by the functions θ(t) and ϕ(t), respectively, where 0≤t≤4 and t is measured in minutes (see figure). These angles are measured in radians, where θ=ϕ=0 represent the starting position and θ=ϕ=2π represent the finish position. The angular velocities of the runners are θ′(t) and ϕ′(t). <IMAGE>
b. Which runner has the greater average angular velocity?

Verified step by step guidance

1. To find the average angular velocity of each runner, we need to use the formula for average angular velocity, which is the total change in angular position divided by the total time taken. Mathematically, this is expressed as: \( \text{Average Angular Velocity} = \frac{\Delta \theta}{\Delta t} \) for Jean and \( \frac{\Delta \phi}{\Delta t} \) for Juan.
2. Determine the total change in angular position for each runner. Since both start at \( \theta = 0 \) and \( \phi = 0 \) and finish at \( \theta = 2\pi \) and \( \phi = 2\pi \), the change in angular position for both runners is \( 2\pi \) radians.
3. The total time for the race is given as \( \Delta t = 4 \) minutes.
4. Calculate the average angular velocity for Jean using the formula: \( \text{Average Angular Velocity for Jean} = \frac{2\pi}{4} \).
5. Calculate the average angular velocity for Juan using the formula: \( \text{Average Angular Velocity for Juan} = \frac{2\pi}{4} \). Compare the two values to determine which runner has the greater average angular velocity.

Key Concepts

Angular Position

Angular position refers to the angle that an object has rotated from a reference point, typically measured in radians. In the context of the race, θ(t) and ϕ(t) represent the angular positions of Jean and Juan over time, indicating their respective locations on the circular track. Understanding angular position is crucial for determining how far each runner has traveled around the track.

Angular Velocity

Angular velocity is a measure of how quickly an object rotates around a point, expressed as the rate of change of angular position with respect to time. It is denoted as θ′(t) for Jean and ϕ′(t) for Juan. To compare the average angular velocities of the two runners, one must calculate the change in their angular positions over the time interval, which provides insight into their speed around the track.

Average Angular Velocity

Average angular velocity is defined as the total change in angular position divided by the total time taken. For the runners, it can be calculated using the formula (θ(final) - θ(initial)) / (t(final) - t(initial)) for Jean and similarly for Juan. This concept is essential for determining which runner has a greater average angular velocity over the course of the race, allowing for a direct comparison of their performances.