A race Jean and Juan run a one-lap race on a circular track. T...

Question

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?

b. Which runner has the greater average angular velocity?

Accepted Answer

Welcome back, everyone. Two cyclists A and B are racing on a circular track. Their angular positions during the race are given by the functions alpha of T4A and beta of T4B. Where T is time in seconds and time is between 0 and 10 inclusive, at time of 0, both cyclists are at the starting line and at time of 10 they have both completed one full circle. Which cyclist has the greater average angular velocity were also given the sketch. The sketch illustrates the alp of T and beta of T functions. We can see that at time of 10 they both completed a full circle, and their angular position is the same. So what we want to do is simply recall that the average angular velocity omega average is defined by the change in angular position delta theta. Divided by the change in time delta t so we can say that. A And B, they both complete a full circle, which essentially means that their final angular position is 2 pi. They both begin at 0, right? So 2 minus 0 gives us delta theta equals 2 minus 0, which is 2 by radiance. What about the change in time? Well, once again, both for A and B, the change in time would be equal to 10 minus 0, which is 10 seconds. Right? Essentially we can see that from the graph as well. So they complete one full circle in the same period of time. So regardless what the functions alpha and beta are, they will have the same angular velocity. Omega average is equal to 2 by radiance. Divided by 10 seconds, right, so those parameters are the same for each cyclist. We can of course evaluate the result if we want. We're going to get pi divided by 5 gradients per second for each. So now we can conclude that they have. The same Angular Velocity, which is our final answer. Let's label it and thank you for watching.

Key Concepts

Angular Position

Angular Velocity

Average Angular Velocity

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