A circular track has several concentric rings where people can run at their leisure. Phil runs on the outermost track with radius rP while Annie runs on an inner track with radius rA = 0.76 rP. The runners start side by side, along a radial line, and run at the same speed in a counterclockwise direction.
A.) How many revolutions has Annie made when Annie's and Phil's velocity vectors point in opposite directions for the first time?
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If V is the speed that Phil and Annie run, θ is the angular distance they run, ω is their angular velocity, and t is time, then:
θp = ωp t = (V / rp) t
θa = ωa t = (V / ra) t
The first time their velocity vectors point in opposite directions, θa = θp + π.
θa = θp + π
(V / ra) t = (V / rp) t + π
(V / ra) t - (V / rp) t = π
V t (1/ra - 1/rp) = π
V t (1/ra - 1/(ra/0.76)) = π
V t (1/ra - 0.76/ra) = π
V t (0.24/ra) = π
0.24 V t / ra = π
Number of revolutions is:
θa / (2π) = ((V / ra) t) / (2π)
N = (V t) / (2π ra)
Rewriting:
2π N = V t / ra
Substituting:
0.24 (2π N) = π
0.48 N = 1
N = 2.083
Annie has run 2.083 revolutions when the velocity vectors point in opposite directions for the first time.