Time To Go⚠ unverified

Understanding the Parameters

• Ground speed, not airspeed — the crucial distinction — an aircraft moves through the air at its airspeed, but the air itself moves over the ground (wind). Time to go depends on progress over the ground, so it must use ground speed = airspeed combined with the wind vector. A 500-kt airliner into a 100-kt headwind makes good only 400 kt over the ground, and its time to go is 25 % longer than airspeed alone would suggest. GPS measures ground speed directly; without GPS it comes from the wind triangle.

• Distance remaining $d$ — usually the great-circle distance to the active waypoint, continuously recomputed as the aircraft moves. In a flight plan it is the along-track distance to the next fix.

• From time-to-go to ETA — add $t$ to the current time and you have the ETA; sum the times-to-go over all remaining legs and you have the arrival time at the destination. Multiply by fuel flow and you have fuel required. This one quotient propagates into the entire flight-management picture.

• The $V_g \le 0$ guard — if ground speed is zero or negative (holding, or blown backward by an extreme headwind), there is no finite arrival time, so the calculator returns infinity rather than a negative or divide-by-zero result. Physically: you are not getting there.

• Constant-speed assumption — the formula assumes a constant ground speed over the remaining distance. Real ETAs integrate a changing speed profile (climb, cruise, descent, changing winds), so flight computers sum time-to-go over many short segments — but each segment is this same $d/V_g$.

Derivation (Approaching a Proof)

This is the definition of average speed, solved for time. Ground speed is distance covered over the ground per unit time:

$$V_g = \frac{d}{t}.$$

For motion at constant ground speed, invert to get the time to cover the remaining distance $d$:

$$t = \frac{d}{V_g}. \qquad\blacksquare$$

For a varying ground speed the exact statement is the integral $t = \int_0^d \dfrac{ds}{V_g(s)}$, which reduces to $d/V_g$ when $V_g$ is constant; flight-management systems evaluate it piecewise over the route. The $+\infty$ result for $V_g\le0$ is the limit of $d/V_g$ as $V_g\to0^+$.

Dimensional check. $\dfrac{d}{V_g} = \dfrac{\text{m}}{\text{m}/\text{s}} = \text{s}$ ✓ — a time.

History and Development

• Dead reckoning. "Distance over speed equals time" is the arithmetic core of dead reckoning, the centuries-old navigation method of computing position from heading, speed, and elapsed time. Every pre-GPS navigator's log was built on this relation applied leg by leg.

• The E6B and the flight computer. The circular slide-rule E6B (1930s, Philip Dalton) mechanised time–speed–distance and wind-triangle problems for pilots; "time to go" was one of its staple computations, set by aligning speed against distance on the rotating scales.

• Flight management systems. Modern FMS compute time-to-go and ETA to every waypoint continuously, feeding required-time-of-arrival (RTA) functions that let air-traffic control meter aircraft to a fix within seconds — the same $d/V_g$, evaluated thousands of times a second with GPS ground speed and integrated wind and speed profiles.

Related Concepts: Great-Circle Distance, Initial Heading, Cross Track Error, Ground Track Velocity, Specific Range, Range Breguet

Notes: Exact definition $t=d/V_g$; ETA = clock $+ t$. Must use ground speed (airspeed ± wind), not airspeed — GPS gives it directly. Returns $+\infty$ if $V_g\le0$. Constant-speed assumption; real ETAs integrate $\int ds/V_g(s)$ piecewise over the route. The arithmetic core of dead reckoning and the FMS. Defaults ⇒ $t=4\ \text{s}$.