DIY · Card 32 · Engineering · Flight
A strip of paper, three snips and a paper clip. Drop it from up high and it doesn’t fall: it whirls, spinning its way gently to the floor like a sycamore seed.
01The brief
Make a paper helicopter that spins, and takes as long as possible to reach the floor.
1 m/s
about how fast a good paper helicopter falls, slower than a gentle walk
5
turns a second, roughly, for a copter with 8 cm blades
2021
the year a helicopter first flew on another planet, Mars
Drop a strip of paper and it flutters down in a second. Cut it, fold it and add a paper clip, and the same paper becomes a helicopter that spins and floats down for two or three seconds. Nothing has been added but a shape.
This card is about that shape, and about a way of working. A paper helicopter is quick to make, so you can make ten, change one thing on each, and find out what really helps. Engineers call that experimenting, and it is the same thinking as tuning the balloon car or catching the falling egg in the inertia egg drop.
02Kit
Paper and paper clips, the only materials allowed, and a pair of scissors.
Tick off what you have
0 of 4 ready
03Method
Cut a strip, make three cuts, fold the body, add a clip, fold the blades opposite ways and drop it.
Step 1 of 8 · Minute 0
You need a sheet of paper, a ruler, a pencil, scissors and a few paper clips. That’s all: the challenge is to make it fly with paper and clips alone.
It falls straight down without spinning.
Why
The blades are folded the same way, or not folded down far enough.
Fix
Fold one blade forwards and the other backwards, both right out to the side.
It tumbles over and over.
Why
Not enough weight at the bottom, or the blades are uneven.
Fix
Add a paper clip, and trim the blades so they are exactly the same length.
It spins but drops fast.
Why
Too heavy, or the blades are too short to catch much air.
Fix
Use one clip, make the blades a little longer, and drop it from higher up.
04Physics
As soon as you let go, gravity pulls the helicopter down and air rushes up past it. Each blade is folded out at an angle, so that air is pushed sideways off it, and the blade is pushed the other way. Because the two blades are folded in opposite directions, the two sideways pushes go round the same way, and the rotor turns.
Once it spins, the two blades sweep out a whole disc instead of two thin strips, and that disc behaves like a little parachute. The helicopter speeds up only until the drag of its disc matches its weight, then falls at that steady speed, its terminal velocity. Change the design and drop it:
Fall time
2.28 s
Fall speed
1.10 m/s
Spin
5.2 turns/s
Without spinning
0.73 s
Spinning, the two blades sweep out a whole disc of air, which behaves like a small parachute. The same strip with its blades up hardly slows down at all.
In 1992 the statistician George Box published a short paper about teaching engineers with paper helicopters. His point was that you can learn about two changes at once, if you plan the tests. Make four designs, every combination of two blade lengths and one or two clips, and drop each a few times. Each effect is then the average of half the drops minus the average of the other half. Run it:
| 1 paper clip | 2 paper clips | |
|---|---|---|
| 7 cm blades | no drops yet | no drops yet |
| 11 cm blades | no drops yet | no drops yet |
Effect of longer blades
—
Effect of a second clip
—
Drop each of the four designs at least once. Every drop is timed with a little stopwatch wobble, just like the real thing.
Both widgets run on a simple model of a spinning disc falling at its terminal speed, tuned to behave like real 80 g/m² paper copters. The stopwatch in the experiment adds a random few percent to every drop, as real timing does.
05The big idea
Spin to
slow down.
Two thin blades on their own catch almost no air. Set spinning, they sweep a disc many times their own area, and the air has to push against all of it. Spinning turns a little wing into a big one.
Seeds discovered this long before people did, and helicopter pilots trust their lives to it when an engine stops. The energy of the fall goes into the spin, and the spin holds the fall back.
06The math
1 · Drag
D = ½ ρ C A v²
The push of the air on a disc of area A moving at speed v through air of density ρ. C is a drag coefficient, near 1.
2 · Terminal speed
v = √(2 m g ⁄ ρ C A)
It stops speeding up when the drag equals its weight mg. Solve for v.
3 · The disc
A = π R²
Blades of length R sweep a circle of this area, far more than the paper itself.
A strip 5 cm by 20 cm of printer paper weighs about 0.8 g; add a 0.5 g clip and the copter is 1.3 g. Blades 8 cm long sweep π × 0.08² ≈ 0.02 m². With C ≈ 0.9, its terminal speed is √(2 × 0.0013 × 9.8 ⁄ (1.2 × 0.9 × 0.02)) ≈ 1.1 m/s, so it takes about 2.3 seconds to fall 2.5 metres.
The formula also tells you what to change. Speed goes as the square root of mass, so halving the weight only slows it by about 30 %. But it goes inversely with blade length, because area grows with R²: blades a third longer fall a quarter slower, as long as they stay stiff.
07Real life
Seeds, rescue helicopters and a robot on Mars all depend on the air turning a rotor.
Sycamore and maple seeds
The winged seeds of sycamores and maples are nature’s paper helicopters. They spin as they fall, staying aloft long enough for the wind to carry them away from the parent tree. A 2009 study found that a small whirlwind over each wing’s front edge gives them extra lift.
Helicopters in an emergency
If a real helicopter’s engine fails, the pilot can still land. They tilt the blades so that air rushing up through the rotor keeps it spinning, then use that stored spin to cushion the landing. Pilots practise this autorotation as part of their training.
Mars helicopter
In April 2021, NASA’s Ingenuity became the first aircraft to make a powered, controlled flight on another planet. Mars’s air is about a hundredth as dense as Earth’s, so its long blades had to spin at around 2,500 revolutions a minute to lift its 1.8 kg.
Autogyros
In 1923 the Spanish engineer Juan de la Cierva flew the first successful autogyro, an aircraft with an unpowered rotor on top and an ordinary propeller at the front. Air flowing through the rotor spun it, exactly as it spins your paper copter, and the rotor held it up.
Leonardo’s aerial screw
Around 1489 Leonardo da Vinci sketched a machine with a great spiral of linen meant to screw itself up into the air. It could never have flown, but it shows how long people have dreamed of lifting off with a spinning wing.
Bamboo-copters
Children in China have played for centuries with the bamboo-copter: a little propeller on a stick that flies up when you spin the stick between your hands. It is the paper helicopter in reverse, with your hands doing the spinning and the spin making the lift.
2021
On 19 April 2021, in Jezero Crater on Mars, a 1.8 kg helicopter called Ingenuity spun up its two pairs of blades, lifted three metres off the ground, hovered for about 40 seconds and landed. It was the first powered, controlled flight on another world.
It was meant to fly five times. It flew 72, until a damaged blade grounded it in January 2024. Its engineers faced your problem in an extreme form: air so thin that a rotor has to be large, light and very fast to catch enough of it. A small piece of fabric from the Wright brothers’ 1903 Flyer travelled with it.
08Is and isn’t
The blades push air down, like a real helicopter’s.
There’s no engine, so nothing pushes. The falling copter meets air rushing up past it, and that air turns the blades, like wind turning a windmill. A powered helicopter does the opposite: its engine spins the blades to force air down.
A heavier helicopter spins more slowly.
It spins faster. Extra weight makes it fall faster, so more air rushes past the blades each second and turns them quicker. It just reaches the floor sooner.
The longer the blades, the slower it falls.
Only up to a point. Long, narrow paper blades flop, flutter and stop holding their angle, and then the copter tumbles. There is a best length for each width of strip.
One drop tells you whether a design is better.
Drops vary by a few percent: a draught, a wobble, a slow thumb on the stopwatch. To be sure a change helped, time several drops of each design and compare the averages.
09Go further
A
Ten-drop average
Drop the same copter ten times from the same height and time each one. How much do the times vary? What is the average?
B
Blade length
Make copters with blades 4, 6, 8, 10 and 12 cm long. Plot fall time against blade length. Where is the best length?
C
Clockwise or anticlockwise
Swap which blade folds forward. Does it spin the other way? Does it fall any differently?
D
Seed hunt
Collect sycamore or maple seeds and drop them beside your copter. Which stays up longer, and how do their shapes differ?
E
Box’s experiment
Change two things at once in a planned grid of four designs, like the experiment above. Can you find the effect of each change with only four kinds of copter?
10Questions
Cut a strip of paper about 5 cm wide and 20 cm long. Cut down the middle of the top 8 cm to make two blades, make a short cut in from each side a little lower down, and fold the sides of the body in. Fold up the bottom, hold it with a paper clip, then fold one blade forwards and one backwards, and drop it from high up.
Because its two blades are folded in opposite directions. As it falls, air rushes up past the blades and is deflected sideways by each of them, pushing one blade one way and the other the opposite way. Together those pushes turn the rotor, the same way wind turns a windmill.
Make the disc its blades sweep bigger and the helicopter lighter. Longer blades help, up to the point where they flop; wider blades stop the flopping. Use only one small paper clip, and make sure both blades are folded to the same angle so it spins smoothly.
Autorotation is when a rotor is spun by the air flowing through it, rather than by an engine. A paper helicopter, a sycamore seed and a real helicopter whose engine has failed all use it: as they fall, air spins the rotor, and the spinning rotor slows the fall.
How to run a fair experiment. Because every drop is a little different, you time several drops of each design and compare averages, changing one or two things at a time. The statistician George Box used paper helicopters for exactly that, to teach engineers experimental design.
Safety · Drop helicopters from the top of the stairs or with an arm stretched up, never by leaning over a banister or standing on a wobbly chair. A grown-up should supervise any high drops.
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