Departure Facts - A Short Course in Basic Physics

by H. Clay Gorton

There are a number of obvious generalities with which we are all familiar about how to get an airplane in the air. We know, for instance, that take-off distance is increased by decreased thrust, increased drag, higher weight and decreased headwind–but by how much? After starting a take-off roll it may be necessary to make a split-second decision to abort if any or a combi­nation of the above factors get out of whack. That split-second is not the time to review generalities in order to make a life-saving decision. However, with specific knowl­edge of the magnitude of the various factors on the distance from brake-release to lift off, we can look down the runway and make a life-saving decision before we get into a critical situation.

Let’s look at the specifics of the effect of take-off distance on acceleration, weight and wind velocity.

Acceleration. Although we may apply a constant maximum power during the take-off roll, acceleration is not constant. Two opposite effects on drag occur during lift off. Rolling friction decreases as wing lift increases with velocity, but airframe drag increases with velocity. Further, propeller thrust decreases with increased velocity. The sum of all these factors tends to decrease the rate of acceleration during the take-off roll. However, to calculate specifics we will assume a constant acceleration, with caution that the numbers will be somewhat optimistic.

The distance, D, required to reach take-off speed may be written as D = V2/2a, where V is the aircraft speed and a is its acceleration. If the lift-off speed is 60 knots (100 ft/sec) and you accelerate, let’s say, at 0.1g, or 3.2 ft/sec2, the take-off distance will be 1562 feet.

The time, t, from brake release to lift off may be expressed as t = V/a. For the same conditions of lift-off speed and acceleration, the time would be 31 seconds.

Now let’s look at the time and distance at half the lift-off speed. The time will be 15.5 seconds, but, from the first equation above, the distance will be only 390 feet! In other words, in half the time to lift off, we will have covered only one fourth the distance. At half the lift-off distance our speed will be 70 ft/sec, or 42 knots. In other words, if you haven’t reached three fourths of your lift-off speed in half the runway, ABORT!

Next let’s examine how these numbers are affected by weight and wind speed.

Weight. The effect of increased weight on lift-off speed may be expressed as V(heavy) + V(light) {W(heavy/W(light)}½. Let’s assume that our aircraft with a take-off roll of 1562 feet with one occupant, 1/4 tank of fuel and no baggage weighs 2000 pounds. We will increment the weight of added fuel, passengers and baggage in steps of 250 pounds. The following table will show the effect of added weight on take-off speed; and from the first equation we will show the effect of added take-off speed on the distance to lift-off.

Weight Speed (knots) Distance (feet)
2000 60 1562
2250 (12.5%) 64 (7%) 1758 (12.5%)
2500 (25%) 67 (12%) 1953 (25%)
2750 (37.5%) 70 (17%) 2148 (37.5%)

Thus, the increase in take-off distance is proportional to the increase in gross weight under standard conditions. Taking into effect runway temperature, the take-off distance would be further increased by 15% for every 10°F rise in temperature. Thus, if the take-off distance of 1562 feet at a gross weight of 2000 pounds were the case for standard temperature (59°F), at a temperature of 90°F, for instance, the lift-off distances shown above would all need to be increased by 45%. The take-off distance for 750 pounds added weight at 90° is increased from 2148 feet to 3115 feet, which is twice the distance for the single occupant, no load case at standard temperature!

Wind Speed. The headwind or tailwind component of the surface wind has a very large effect on take-off roll distance. The hazardous condition, of course, is to take off or land downwind. The equation for the effect of tailwind on take-off roll distance may be expressed as Dw = Do (1 + vw/V)2, where Do is the take-off distance with no wind, vw is the headwind velocity and V is the lift-off speed. (To calculate take-off roll distance into a headwind, change the sign before the vw to a minus). The table below shows the effect of tailwind and headwind velocity on take-off distance for a lift-off speed of 60 knots and a no-wind roll distance of 1562 feet.

Wind Speed
(knots)
Downwind Roll
Distance (feet)
Headwind Roll
Distance (feet)
5 1833 (15%) 1350 (13%)
10 2126 (36%) 1084 (30%)
15 2440 (56%) 878 (43%)

Note that tailwind has a stronger effect on take-off distance than headwind, and that a tailwind component of only 10 knots will increase take-off distance by over one third!

Working the above equations for your own aircraft could save your life and the lives of your passengers.