#### 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.