## Physics for Scientists and Engineers: A Strategic Approach with Modern Physics (4th Edition)

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.