Question

A circular track has several concentric rings where people can run at their leisure. Phil runs on the outermost track with radius r_P while Annie runs on an inner track with radius r_A = 0.80r_P. The runners start side by side, along a radial line, and run at the same speed in a counterclockwise direction. How many revolutions has Annie made when Annie’s and Phil’s velocity vectors point in opposite directions for the first time?

Short answer

The number of revolutions Annie has completed before both the runners velocity vectors are opposite to each other is 2

Step-by-step solution

Write the given information

Phil runs on the track of radius= r_P
Annie runs on the track of radius = r_A=0.80r_P

To determine the number of revolutions completed by Annie when both the runners have velocity vectors opposite to each other

Let the speed of both the runner is v
The distance covered by Phil in one revolution = 2πr_P
The distance covered by Annie in one revolution = 2πr_A
The angular distance covered by Phil in time t
${\theta }_P=\frac{vt}{r_P}$

Similarly, the angular distance covered by Annie in time t
${\theta }_A=\frac{vt}{r_A}$

At the time, when the velocity vector of both the players is opposite
$$\begin{gathered} {\theta }_A={\theta }_P+\pi \\ \frac{vt}{r_A}=\frac{vt}{r_P}+\pi \\ \left(\frac{1}{r_A}-\frac{1}{r_P}\right)vt=\pi \\ \left(\frac{r_P-r_A}{r_P{r}_A}\right)vt=\pi \\ \left(\frac{r_P-0.8r_P}{r_P\left(0.8r_P\right)}\right)vt=\pi \\ \left(\frac{0.2r_P}{0.8{r_P}^2}\right)vt=\pi \\ \frac{vt}{4r_P}=\pi \\ vt=4r_P\pi \\ vt=2\left(2r_P\pi \right) \\ \frac{vt}{2\pi r_P}=2 \end{gathered}$$

Thus, the number of revolutions Annie has completed before both the runners velocity vectors are opposite to each other is 2