Pick a scenario; every slider below stays editable.
Grey: a lone goose. Gold: flock average. Dashed lines mark the minimum-power and maximum-range speeds of a lone bird; the white dot is your airspeed.
Bird 1 is the leader. Bars below 100 % mean that bird rides upwash from the birds ahead. Without leader rotation the leader sets the pace of the whole flock.
The science
Lift needs downwash. A wing holds weight L = m g by pushing air down. For an elliptically loaded wing of span b the induced drag is Di = k * 2 L² / (pi rho V² b²); k = 1.2 covers non-ideal loading (Pennycuick-style factor).
Wake model. Each bird sheds two trailing vortices a distance b' = (pi/4) b apart with circulation Gamma = L / (rho V b'). At lateral offset y, a distance x behind, the vertical air speed is the sum over the two vortices of +/- Gamma/(2 pi) * d/(d²+rc²) * (1 + x/sqrt(x²+d²+rc²))/2 with d = y -/+ b'/2 (core rc = 0.12 b', semi-infinite filaments). Outside the vortices the air goes up; a follower with its wingtips there averages that upwash over its elliptic lift distribution and its induced drag becomes k L (wself - wup) / V.
Power. Pmech = (Di + Dbody + Dprofile) V, with body drag 1/2 rho V² CdA and wing profile drag 1/2 rho V² S CD,pro. Only the induced part is reduced by the formation. Chemical power is Pmech / 0.23 and fat holds about 37.6 MJ/kg. The lone-bird curve is P = a/V + c V³, so the minimum-power speed is (a/3c)1/4 and the best-range speed (a/c)1/4 = 31/4 Vmp.
Wind. With tailwind u and crosswind w the bird crabs into the wind and its ground speed is sqrt(V² - w²) + u. Power depends on airspeed only, so a tailwind buys range and a crosswind costs it.
Honest simplifications. Steady, non-flapping aerodynamics (flapping unsteadiness and wake phase effects are ignored); the wake does not roll up, decay or spread; birds hold position exactly; upwash is averaged over the span; no thermals or turbulence. Bird constants (3.1 kg lean mass, 1.5 m span, 0.25 m² wing area, body CdA 0.0019 m², profile CD 0.014, efficiency 0.23) are typical illustrative values, not measurements of one goose. Lissaman and Shollenberger (1970) reported roughly 70 % induced-drag reduction for a 25-bird V under comparable assumptions; real flocks save less and the benefit is still debated. The landscape, route and colours are decorative.