The physics of flight paths


In my ongoing experiment to see how technical I can make these posts without people getting annoyed, this is a new frontier. Not for the faint-hearted today!

When you throw a long disc, with a lot of pace, you need to start it more inside-out than a shorter throw. (You doubtless know that already.) At high speeds, the disc tries to flip towards an outside-in angle, so you need to start the disc with more I/O – sometimes nearly vertical for really big throws.

And yet, at the end of the flight, that same throw will die off to the left (for a right handed backhand) – exactly the opposite behaviour. It turns one way at high speed, and the other way at low speed. This is very odd.

And much more difficult to explain than you might think. Fortunately, I know you all love a bit of physics…

It’s not easy to come up with possible explanations – even wrong ones. It’s pretty counter-intuitive when you think about it that a disc should turn different ways at different speeds.

Obviously the disc is symmetric, so that doesn’t help us. There IS an asymmetry in its movement – one side is spinning towards where it’s going and one side is spinning away – so perhaps this means there’s more lift on one side than the other. But why would that swap over at low speed? We might be able to explain the disc tipping one way, but there’s no way that idea can explain it tilting back the other way later on. Whatever made it turn over is surely still happening. It might be happening a bit less strongly as the disc slows, but it should surely still be pushing in the same direction.

Fortunately, Bill Nye to the rescue, demonstrating a key concept we’ll need:

There are three concepts involved in our overall explanation. First is the reaction-at-a-right-angle thing that is demonstrated in the video. Second, we need to think about the lift on the disc. And third, we need to think about the angle of attack of the disc to the air it travels through.

The angle of attack is how much the front edge of the disc is raised – how much ‘stall’ is on the disc. A completely flat disc has a zero angle of attack. A disc with the front edge up, with a bit of touch and stall on it, will have a positive angle of attack – somewhere from 0-20° probably.

But that’s the angle relative to the ground – not to the air. A disc that is horizontal may well start with a 0° angle of attack to the air, but at the end, as it slows and falls, this will have changed. Even if it’s still flat in relation to the ground, it is now travelling downwards, and so its angle of attack to the air has gone way up. A disc that loses all its forward speed and ends up dropping straight down at the end would have as much as a 90° angle of attack – it’s hitting the air with the full flat underside of the disc even though it’s still horizontal (to the ground).

Presentation2

So the angle of attack is almost always low at the start of the flight¹ and much, much higher at the end as the disc starts to fall to earth.

Why does any of that matter? Well, it turns out that the lift force on the disc – the force that keeps it in the air longer than a brick – does not always act in the centre of the disc but toward the front or back. And it further turns out that the determining factor for where that force acts is the angle of attack². When the angle of attack is less than 9°, the lift acts behind the middle of the disc – toward the back.

Let’s think through the consequences of that. Gravity can be assumed to act at the centre of mass of the disc – which for a symmetric single-material object is going to be in the dead centre. So the centre of the disc is being pulled down – but somewhere towards the back of the disc, the lift is pushing up.

If we push down in one place and up in another, the disc should twist – that’s obvious. If we push the back of the disc up while the middle is pulled down, then the front of the disc should tip down. It should nosedive (and actually, it does, when there’s no spin on it.).

But spin changes everything, just like Bill was telling us. The spinning object makes the force act at 90° to where it normally would, and so instead of twisting from front to back as you might expect, the disc twists left to right. What started I/O will straighten; what started flat will flip over³.

And when the angle of attack is more than 9° – either because it’s thrown with a lot of stall, or because it has started to fall to earth – the lift force acts towards the front of the disc. By the same logic as before, the throw will now die off in the opposite direction. The angle of attack late in the flight is almost always much higher than 9°, as the disc loses forward momentum and drops down to the ground, and so it will die off.

So – now you know. It isn’t actually the speed of the disc that influences which way it tilts in flight, but the angle of attack. This information seems close to useless (although fun) – but it does have some implications for gameplay. It means that the more stall you put on a throw, the less I/O you will need.

Your high hucks will die left; your low hucks will flip right. It doesn’t mean you threw it with too much or too little I/O – it means you threw it with too much or too little for the amount of stall you put on it. The height of a huck affects how much it will turn, and you need to compensate with the right amount of I/O to deal with that.

And now you know why those very short passes need to start outside-in. If you throw it so gently that it immediately starts to fall to earth, then it will never have a ‘flipping’ phase; it will only die off. It will start with a steep angle of attack, and you need to allow for that in the throw you make.

I don’t imagine many people will learn anything from this which changes how they play or coach – but I don’t care. I love this stuff… 🙂

1 Even a disc thrown upwards with the front edge raised will have a low angle of attack early on – it’s angled up, but it’s also travelling up early in the throw so the angle of attack to the air is low. Only air-bounce throws have a really sizeable angle of attack early in the throw (or very soft throws – meaning very little lift – which immediately start falling to earth).
2 This is really complicated! You can if you like just accept it as fact. It may just be a complex result of the way the airflow travels over the disc – some nasty equations in fluid dynamics that we really don’t want to get into.
But one thing I like to think about is that there are two types of lift on the disc – one caused by the ‘wing’ shape of the disc (i.e. the air has further to travel over the curved top than the flat bottom, meaning lower air pressure above the disc and thus some lift via the Bernoulli effect) and one caused just by the angle of attack (e.g. if you hold your hand out of a car window and angle it to the wind, it will feel lift just because it hits the air at an angle – nothing to do with wing shapes!).
With a flat disc – a 0° angle of attack – there’s none of that car-window lift at all, so all the lift is from the disc acting like a wing. Let’s assume that acts toward the back of the disc. If we assume that the car-window effect acts towards the front of the disc, and increases with the angle of attack, then at some point (which turns out to be above 9° for an Ultrastar) that lift will be larger than the Bernoulli lift, and the average position of the lift force will be towards the front of the disc.
That’s a possible explanation – no idea if it would be right though!
3 At higher speeds, there’s more lift and hence there’s a stronger force trying to twist the disc. Longer, faster throws will flip more than shorter, softer ones with the same angle of attack – which is why you need to start your hucks inside out.
You can fight it to some extent though – having more spin makes it harder for the disc to be twisted (due to the angular momentum), so more spin means the disc will hold its line better, both against the flipping force early on and the dying off force at the end. Discs with faster rotation fly straighter for longer.
4 The only way to prevent the angle of attack rising during the throw (and hence the disc trying to die off left) is to throw the disc downwards – front edge down – so that the throw never loses forward momentum (since gravity helps to accelerate it and fight the drag). This is a pretty useless option in Ultimate as the ground tends to get in the way – though it can be done on a downhill disc-golf hole. It’s possible to approximate what frisbee-scientists would call a ‘downward steady glide’ – a straight and unchanging flight. Like I say though, that can’t be done in Ultimate… your throws will all change orientation at some point.
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14 Responses to The physics of flight paths

  1. Hildo Bijl says:

    It’s a nice explanation you’ve put together. It’s awesome how you can use scientific knowledge to explain stuff which most Ultimate players actually already know. However, the real die-hard geeks already knew everything in this post. So for those people, I would like to offer a follow-up question.
    I’ve noticed that the type of wind also affects the behavior of the disc. When there is a very strong but steady wind, discs behave pretty much as normal. During normal flight, they go slightly from inside to outside. However, when there is a very irregular wind, with lots of turbulence, vortices and all that, discs transition from inside to outside a lot faster, even when the wind speed is the same. Why is that? (Or to use scientific terms: why would a strongly varying/stochastic wind distribution result in an additional shift of the center of pressure to the rear of the disc?) If I have some extra spare time, I’ll see if I can come up with some theories myself too.

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    • Followup thoughts to the wind question – when a disc pops up, it typically drops dramatically shortly afterwards. My throwing buddy and I have observed that most of the time, if you throw a flat backhand with good rotation to a receiver’s chest, it tends to wind up at the desired height even if it hops up (assuming a gust doesn’t blow it offcourse) because the disc is probably going to take just as dramatic a plummet before it gets there.

      But this seems to go out the window if you leave a disc up above head level. That’s when stuff gets messy and the wind really gets to the disc. Obviously the wind is faster at six feet off the ground than it is at three feet (less drag effect on the fluid), but it also seems that new mold discs seem to normalize at a higher flight than old mold discs – all of our throws (and we have very different throwing motions) tend to wind up at shoulder/head level instead of just above the waist. So I’m not sure if it’s the aerodynamics of the different molds that plays a role – I’ve come to believe it’s harder to control an old mold disc but you can do more with it – or if it’s just the fact that wind plays more of a factor a few more feet off the ground.

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      • Crumbs – not a clue. I can’t say I’ve noticed these behaviours. Here’s an idea though. Let’s say the wind is 5mph at waist high (gusting 0 to 10) and 10mph overhead (gusting 5-15). It’s possible that a lower disc will go up and down, but a higher disc will go up or remain the same height – the wind is never low enough for the disc to actually drop, it just goes along ‘normally’ in between the leaps.

        In general, I wouldn’t be surprised to find that the majority of discs end up about where you aim them. Whatever the average wind is, you probably took that into account, and the odds are that when it’s all swirly and confused then most of the time any patches of higher pressure will have pockets of lower pressure behind them, so on average it’ll be about what you expected over any meaningful distance. Pretty much just regression to the mean. The chances of hitting 5 high pressure areas and no low pressure areas is just lower. As for being different higher up, well, I’m only guessing on that…

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    • Hildo – I can’t explain it in full, but I can suggest a reason why things are different in a gusty wind. Look here (and apologies for anyone who doesn’t like swearing…) http://goo.gl/a0Et1c

      It’s a very similar question to “Why does a beat-up golf disc turn over more?” or “Why do wobbling throws turn over more?”. The answer I think has to come from the changed airflow over the disc.

      The flight rings set up changes in the boundary layer that create a smooth area, as you see in that linked article. I can easily imagine that turbulent air would make a big difference to the way that air adheres to the disc, changing (overriding?) the smoothing influence of the flight rings and making the disc behave differently. Whether it’s the wind or the wobble or the rough edge changing the airflow, I guess it’s not that big a surprise that the disc doesn’t fly quite the same.

      Why it specifically moves the lift further back I couldn’t say, I’m no expert on fluid dynamics. But I think I can see why it would behave a little differently in some way or other.

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  2. Anonymous says:

    Very interesting article, thanks for writing! I understand the basic Bernoulli argument about why Ultimate discs tend to turn outside in due the spin direction, less pressure above the outside forward spinning edge and all. But why is it that disc golf discs do the opposite (turn inside out)?

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    • They don’t. They just turn at different speeds. A Stratus will turn outside-in at a much lower speed than a Firebird, for example. All of them though will turn outside-in if thrown hard enough. And all of them will die off at the end (including of course the Ultrastar, and the Stratus).

      The different shape of the discs, the different edges they have, affect where the centre of lift is. The exact angle required for a disc to turn one way or the other will be different to an Ultrastar, and different to each other. It wouldn’t surprise me if some discs would only turn over with a negative angle of attack (i.e. flying basically horizontal but getting enough lift to actually be rising rather than falling). But they’ll all turn over if thrown hard enough.

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  3. David says:

    Nice explanation, thanks. Connecting the rear acting lift force and gyroscopic motion to the O/I tendency is a nice find! Regarding point ‘2’ though, it turns out most physicists don’t support the ‘Bernoulli lift’ effect being a result of air travel speed.
    bitly.com/1pXpeKg
    OR
    xkcd.com/803/

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    • Dammit – you’re right. Love some XKCD.

      My understanding was that it was to do with air-travel-speed and pressure differences, but that it’s just not about the ‘travelling further’ thing – there are more complicated reasons why the air over the top travels faster. But it does travel faster I thought?

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      • David says:

        Ah, okay. That sounds right. Even elsewhere in the wikipedia article that I cited (see bitly.com/1wFeMuV for example) states that the air over top of the wing does move faster. The fallacy is, just as you stated, that the speed difference results from a ‘equal-transit time’. Thanks!

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  4. BCR says:

    As this is a subject I enjoy i’m torn between saying what a great summary this is and wanting to be a pedant and point out the niggling details. Anyway – I think you and your readers will have some fun trying to apply the same explanations to the more interesting throws and their characteristics (eg Hammers, blades, scoobers etc). Once you add in the rotation around the third axis (eg extreme “IO”) it makes your brain work a little harder – Gravity, Lift, Drag and the momentum of the disc are in different planes and all the stuff that makes a disc so versatile comes out.

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  5. Pingback: Waiting for Irish Ultimate, interview with Raymond Stephens, host of the Glendalough Open | Sarah Paddle Swim

  6. Cut Camp says:

    Great write up of some complicated stuff. I’d love to see a similar write up of the aerodynamics of a hammer throw, especially with the abrupt direction change that can happen on long hammers.

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  7. Francisco says:

    Excellent post, I had been wondering exactly this for some time… I just wish you had been more explicit on why the lift acts on the front or back of the disc and the turning point at 9°… Why is it safe to assume that the “wing effect” (Bernoulli) happens at the back of the disc, and the “car effect” (Newton) happens at the front?
    Also, I am sure you’ve noticed that there is an equilibrium period, in which the disc is not turning to either side, very likely because the net lift force coincides with the centre of mass… Does that mean that the disc is travelling with an angle of attack of 9°? Is the disc then turning until it finds that sweet spot, and then drag pushes it off balance and it starts turning-falling?

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    • On the first point – I have no idea. That seems to be where it acts based on the behaviour, and certainly that was what was found in the Hummel thesis. The exact fluid-dynamical reason why it acts in different places is beyond me for now I’m afraid!

      On the second point – I don’t know. In so far as there is an equilibrium period (and I’m not absolutely sure that there is a prolonged equilibrium that needs explaining, though I agree it does sometimes LOOK as though the disc stays level longer than you might expect if it was just changing smoothly from turning one way to turning the other), then I am unable to explain why the disc would ‘prefer’ to be flat (or alternatively, would ‘prefer’ to have a 9 degree angle of attack). Sorry!

      The best I can do is to suggest that we’re imagining it – basically, anywhere between say 7 and 11 degrees might be close enough to stable that we can’t tell the difference and so it looks like it’s at exactly 9 degrees for a long time. And once we start to drift further away from the ‘stable’ position, the twisting forces increase exponentially, and so we see quite strong turn early and late in the flight but a long, relatively stable portion in the middle. But I agree – it does often seem as though a disc thrown with increased spin simply WANTS to be flat, and if that’s true I don’t have an explanation for it.

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