At a glance
A plane flies 400 km out and 400 km back at a fixed airspeed of 100 km/h through air that may be moving. The sim times that round trip with no wind, wind across the route and wind along it, the same arithmetic as the two arms of the Michelson–Morley experiment.
What you'll learn
- How ground speed relates to airspeed and wind, and which frame measures each.
- How the round-trip time depends on the wind's speed and on its direction.
- How one ratio, β = w/v, governs both windy routes.
- Why this is the calculation behind Michelson and Morley's two arms.
Two airports 400 km apart. The wind is the air's own speed over the ground; the sim offers it along the route or across it.
What's on screen
- Airports A and B (teal): the route's two ends.
- Plane (amber): flies A → B → A at 100 km/h relative to the air.
- Wind (blue arrows): shown in Cases 2 and 3 only.
- Flight readout (top left of the canvas): the current leg, its ground speed, and the hours flown on it. A banner gives the round-trip time on landing.
- Summary and Split timer (sidebar): the total for each case, and live leg times.
See itFly the baseline with no wind. Watch for: the plane at 100 km/h on both legs, and the round-trip timer ending on 8.00 h.
The controls
Press ⌖ on a row to see where that control sits in the sim.
| Control | What it changes | Range | |
|---|---|---|---|
| Case | The wind's direction: 1 no wind, 2 across the route, 3 along the route (tail/head). Choosing a case restarts the flight. | 1, 2, 3 opens at 1 | |
| Wind speed w | The speed of the air over the ground. It has no effect in Case 1, and it restarts the flight. | 0–95 km/h opens at 60 | |
| Fixed numbers | The airspeed and the distance, shown for reference. They never change. | 100 km/h, 400 km | |
| Round-trip summary | The total time of each case at your current w. | panel | |
| Split timer | The outbound, return and round-trip times of the flight on screen, with each leg's ground speed. | panel | |
| ▶ Play / ⏸ Pause | Runs the flight, or freezes it mid-leg so you can read the timer. | top bar | |
| Speed | How fast the flight is replayed. The flight times themselves stay the same. At the opening 0.5×, one hour of flight takes one second. | 0.25×–4× opens at 0.5× | |
| ↻ Reset | Restores w = 60 km/h and 0.5× speed, and restarts the mode you are in from its beginning. | top bar |
Fixed: v = 100 km/h, d = 400 km each way. The wind is steady, and turning round at an airport takes no time.
Tailwind and headwind do not cancel
The engine fixes the airspeed v = 100 km/h, which is the plane's speed in the air's frame. The airports sit in the ground frame, where the air moves at w. The course's rule for a frame moving at V is ũ = u − V; with the air as the moving frame, V = w:
Wind along the route, read in the airports' frame.
- At w = 60 km/h the ground speed is 160 km/h outbound and 40 km/h on the return.
- The legs are equal in distance but not in time: 400/160 = 2.50 h out and 400/40 = 10.00 h back.
- The plane spends four times as long on the slow leg. Over the whole trip its average ground speed is 800 km / 12.50 h = 64 km/h, not the 100 km/h that averaging 160 and 40 suggests.
Case 3, wind along the route: the slow leg sets the total.
Key idea. Equal distances flown at unequal speeds take unequal times. The slow leg lasts longer, so it counts for more.
See itFly the wind along the route at w = 60 km/h. Watch for: the Return timer still counting long after the Outbound timer has stopped at 2.50 h.
A crosswind still costs time
A wind across the route does no work along it, but it pushes the plane off the track. To stay on the line from A to B, the plane points its nose into the wind (the crab angle) so that part of its airspeed cancels the wind's sideways push.
The vectors at w = 60 km/h. The sim draws this mirrored: wind down the screen, nose angled up.
- Only the component √(v² − w²) of the airspeed is left along the route. At w = 60 km/h that is 80 km/h, and the nose points arcsin(w/v) = 36.9° into the wind.
- Both legs run at the same ground speed, so each takes 400/80 = 5.00 h and the round trip 10.00 h.
- That is 2.00 h more than still air. The same wind along the route costs 4.50 h more.
Case 2, wind across the route.
See itFly the wind across the route at w = 60 km/h. Watch for: the nose angled into the wind, and the Outbound timer stopping at 5.00 h with the Return timer following it.
One number, β = w/v, sets both costs
Let β = w/v be the wind as a fraction of the airspeed (the sim keeps it below 0.95). Still air takes t₀ = 2d/v = 8.00 h; each windy route takes t₀ times a factor that depends on β alone:
t∥ is the round trip along the wind, t⊥ the round trip across it.
w
case
These are the sim's own controls.
- The routes tie only at w = 0. Their ratio 1/√(1 − β²) is above 1 for every wind above zero, so the along-route trip is always slower.
- For gentle winds both extra times grow as β²: at w = 30 km/h they are 0.39 h across and 0.79 h along, about 1 : 2.
- As w approaches v the along-route time grows without limit, since the return ground speed v − w goes to zero. At w = 95 km/h: 25.62 h across, 82.05 h along.
See itFly the wind along the route at w = 95 km/h. Watch for: the return leg crawling at 5 km/h, with the plane barely leaving B after ten seconds.
The same sums describe Michelson–Morley
Replace the plane by light, the airspeed v by the speed of light c, and the wind by the aether wind V that the Earth would feel moving through a stationary aether. Then β = V/c, and the two interferometer arms, each of length d, are the two round trips: one along the wind, one across it.
The arm-time difference, with v → c and w → V.
- The instrument has no stopwatch. It recombines the two beams, and a time difference Δt appears as a shift of the interference fringes, ΔN = cΔt/λ for light of wavelength λ.
- For β ≪ 1 the difference is only Δt ≈ (d/c)β², second order in β. Turning the table by 90° swaps the arms, so the experiment looks for a fringe shift as it turns.
- The analogy covers the two round-trip times only. The sim's smallest wind is 1 km/h (β = 0.01), far above the Earth's β = V/c ≈ 10⁻⁴.
See itFly the wind along the route at a gentle w = 5 km/h (β = 0.05). Watch for: the round-trip timer ending on 8.02 h, only 0.02 h above still air.
Where it matters
- Airline round trips. A steady 100 km/h jet stream along a 5000 km route flown at 900 km/h airspeed gives 5.00 h out and 6.25 h back: 11.25 h against 11.11 h in still air. Real winds vary along the route. Pilots call the crab angle the wind-correction angle.
- Michelson and Morley, 1887. About 11 m of folded light path per arm, λ ≈ 590 nm and the Earth's orbital speed of 30 km/s predicted a shift of ≈ 0.37 fringe on turning the table. They measured less than a twentieth of that.
- Light's speed. The null result fits light having the same speed c in every inertial frame, so ũ = u − V does not apply to light.
The apparatus, and the shift that was predicted against the shift that was seen.
Go deeper: the fringe estimate, and the FitzGerald–Lorentz patch
ΔN for one orientation, then for a 90° turn (d = 11 m, λ = 590 nm, β = 10⁻⁴).
- Shortening the along-wind arm to d√(1 − β²) makes t∥ equal t⊥ exactly: the FitzGerald–Lorentz patch.
- In special relativity √(1 − β²) = 1/γ appears as length contraction, measured by an observer relative to whom the object moves, with no aether.
Summary
In one line. Through moving air the round trip is slower than in still air: t₀/(1 − β²) along the wind, t₀/√(1 − β²) across it, equal only at w = 0.
Key points
- Airspeed is measured in the air's frame, ground speed in the airports' frame: v ± w along the route, √(v² − w²) across it.
- Equal distances at unequal speeds take unequal times, so the slow leg counts for more.
- β = w/v alone sets the extra time, which grows as β² for gentle winds.
Common mix-ups
- "A tailwind out and a headwind back cancel." The ground speeds average to v, but the times do not: more time is spent at the slow speed.
- "A crosswind is free." The nose must point into it, leaving only √(v² − w²) along the track.
- "The plane is the interferometer." It is an analogy for the two round-trip times only.
Things to try on your own
- In Case 3, compare the extra time over 8.00 h at w = 30 and w = 60 km/h: the second is far more than double.
- Find the largest whole-number w for which Case 3 takes under 10 h, then check it with the split timer.