Answer
Annie has made 2.5 revolutions when their velocity vectors point in opposite directions for the first time.
Work Step by Step
Annie's velocity vector and Phil's velocity vector will point in opposite directions when Annie has run half a revolution farther around the track than Phil.
We can find the angular speed for each person.
$\omega_P = \frac{v}{R_P}$
$\omega_A = \frac{v}{R_A} = \frac{v}{0.80~R_P} = 1.25~\omega_P$
We can find the total angle through which each person runs in a given time $t$.
$\theta_P = \omega_P~t$
$\theta_A = \omega_A~t = 1.25~\omega_P~t = 1.25~\theta_P$
The difference in the two angles should be 0.5 revolutions.
$\theta_A-\theta_P = 0.5~rev$
$1.25~\theta_P-\theta_P = 0.5~rev$
$\theta_P = 2~rev$
We can find the number of revolutions Annie has made when Phil completes 2 revolutions.
$\theta_A = 1.25~\theta_P = 2.5~rev$
Annie has made 2.5 revolutions when their velocity vectors point in opposite directions for the first time.
