Question for our math professor: Turning point over water?

[yvesg]

I am doing my flight planning for the trip to Oshkosh and there is a possibility I have to cross lake Michigan. If we have the following variables:

Wind at altitude in knots

Wind direction in degrees

Aircraft speed for maximum range in knots

Distance between the two shores in nauticle miles

True course in degrees

Altitude in feet

What is the formula that will allow me to calculate the decision, in time, of the point where it is better to continue across instead of turning back if I have an engine failure?

Am I missing something?

Erik?

Yves

[Bob]

Not big on math here.  I keep it simple, keep an eye on boats and head to one if you have trouble while in the middle of the "Big Pond".  Large boats have, First Aid, communication's, raft's and many people to help.  Just don your vest early and keep it on till you are over land.

 

Something to think about:  I was going from KALO to KLWA at 17.5.  At 65 miles shore to shore, I figured that I would be out of glide for about 2 min, in the middle.   But had a good tail wind and had a 180+ kt ground speed then 220 kt in descend.  It dawned on me that I was out of glide from almost my entire descend that was about 45 miles.  In this case, I figured that I would turn around for the first 3-4 min of descend, then head for a big boat for the rest.

 

You can see my Flightaware log, of N231PZ, from June 13 for details of this example.

[Danb]

Just have a nice life vest by the way lake mich is cold. Look in the aim devices are mandatory

[John Pleisse]

Zero wind, in most places, you have about a 12-14 minute no-man zone. Just shift your no-main zone for the winds. The water is a toasty 61 degrees this time of year, so I wouldn't sweat it. :) :)

[PTK]

One way is to compare times.

This is how I'd calculate it:

The point of equal time is the point where the time required to continue is same as the time required to turn back. Also we understand that in one direction we'll have a headwind component hw, and in the opposite direction a tailwind component, tw.

Assume distance across the lake is D, distance to the point of no return is d, TAS is your TAS, hw is your head wind and tw is tail wind component.

Calculate wind components and set up the appropriate time equations accordingly.

Set them equal to each other:

d/(TAS-hw) = (D-d)/(TAS+tw) OR d/(TAS+tw) = (D-d)/(TAS-hw)

Substitute in the values, x-multiply and solve for d.

Notice that in the theoretical no wind scenario, d is the mid-point of the lake!

Also note that the expressions TAS+tw and -hw are expressions equivalent to ground speed. But in the form I've written them they enable calculation of d in the flight planning stages by entering your TAS and wind components.

Can also take it a step further and take into account our fuel available by doing a similar calculation to compare distance we can fly into lake and be able to turn around and make it back.

Assume t is time into lake and af is available fuel.

t/(TAS+tw) = (af-t)/(TAS-hw) and x-multiply

t(TAS-hw) = (af-t)(TAS+tw)

Solving for t will give the time we can fly into lake on our available fuel and be able to turn around and make it back.

Note again that in theory with no wind we can fly to the mid-point of the lake.

Hope this helps!

[BorealOne]

Nice - d/(TAS-hw) = D-d/(TAS+tw) OR d/(TAS+tw) = D-d/(TAS-hw).

 

Step 2 is calculating the probability of survival (Ps) and multiplying that by d. 

 

d(Ps) > 0 = go for it

d(Ps) = 0 = go around

 

Of course, personal risk tolerances may vary...

[Piloto]

Never though of Lake Michigan to be a risk overflying it. But growing up and learning to fly on an island (Puerto Rico) is common to be overwater on almost any trip (by plane or boat). First time I flew on the continental US i was impressed by how much land you can fly over without seen the ocean. In any case it is wise to carry a life jacket. Better a raft if you can and some form of signaling like flares or a marine VHF radio. A marine radio will call the attention of any boater even if they are not looking in your direction. And you can coordinate your rendezvous with a boat before ditching. Fly high to give you time to prepare and contact for help.

 

José 

[romair]

I would think the point of no return is the point where you would not be within gliding distance anymore, and not necessarily the point where the time required to go forward equals the time to return. It may be way before the halfway point. Not sure yet how to compute it...but you will need to figure out your gliding distance. It will of course vary depending on the wind and direction. Xavion would be helpful in the scenario you're describing.

[29-0363]

I regularly cross water in my M20R (Spencer Gulf in South Australia) where the distance between land is around 55NM. I plan at FL110 or FL120 and I reckon there's only a very short time where an engine failure would give me wet feet. Spencer Gulf water may not be quite as cold as Lake Michigan but it's renowned for its population of Great White sharks.

Something the PNR formula doesn't take into consideration is the distance travelled as the airplane slows from 175K to best glide speed of around 85K while maintaining level flight. Only when you've washed of the all that excess speed and stopped the prop from windmilling would you begin descent. You would hope that extra distance would be the difference between landing in the shallows and landing on the beach.

[BorealOne]

Ah, Aussie-rules flying. I've done a little bit of over-water flying in N.Queensland, along the coast in the NT out of Darwin, and around the Kimberley out of Broome. I figured that I had Buckley's chance either in the shallows (where the salties would eat me) or over the deep (with the sharks) :-)

[yvesg]

I am awaiting for the answer from our math professor lollll before I conclude on this. Thanks for all the suggestions... getting near a boat is definitively the best thing to do if you need to ditch. I have also looked at the US regulations and for small private aircraft, looks like we are not required to carry flotation devices. Just want confirmation. If we plan to go across, I'll have to find the room for some.

Yves

[N201MKTurbo]

It is best to fly over the lake IFR so you can't see the water.

[jlunseth]

I have crossed the Lake west to east, but not in the other direction. What I can add though, is some of the dynamics that I run into every time I fly the Flight Levels and make a descent for landing. The issue is that the math is not linear. My 430 has a function for computing the necessary descent rate to reach a point that you set, say, 10 miles before the airport, and it uses linear math. As an example of what I am talking about, I did a trip last week where if I had followed its direction, and if I wanted to maintain a 500 fps descent rate, I would have started my descent around 200 nm from the airport. And it never works to start a descent that far out, even if the linear math says to. The reason is that both winds and true airspeed change as you descend. Most of the time if you have a great wind aloft, you are going to lose that velocity the closer you are to the earth, but that is not always true. This last trip we actually had more wind in the lower altitudes than high up, but that did not mean we needed to start our descent further out. True airspeed also changes. One issue with the changes in winds aloft is that the forecasts at the various levels are not always very accurate.

The effects of this are smaller if you are making a descent from, say, 8,000 rather than FL210, but the effect is nevertheless there. Computing how far you are going to get in a descent from cruising altitude is as much art as it is math. I always like to have a big "margin of error" when I make an engine out glide range computation and a decision based on that computation, say about 20%.

[BorealOne]

I am awaiting for the answer from our math professor lollll before I conclude on this. Thanks for all the suggestions... getting near a boat is definitively the best thing to do if you need to ditch. I have also looked at the US regulations and for small private aircraft, looks like we are not required to carry flotation devices. Just want confirmation. If we plan to go across, I'll have to find the room for some.

Yves

 

However, as a C-registered plane, you are required to carry floation in accord w/ TC regulations.

[wishboneash]

In my M20J, if I can maintain a climb rate of 760ft/min or better (rate of descent with engine out), I can always return to shore. In this case, assuming L Michigan minimum crossing distance is 45nm, I will have to climb to 12,000 ft to glide to either shore. If I go past half way (22.5nm), I will go to the other side, if not I come back to same side. This is in no-wind condition.

[29-0363]

One also needs to consider that a decision to turn back will take time and corresponding loss of altitude during a 180 degree turn, for no distance benefit. while difficult to establish exactly where that point would be, I would keep heading outbound if the engine became silent about one minute before the PNR.

[aviatoreb]

I am awaiting for the answer from our math professor lollll before I conclude on this. Thanks for all the suggestions... getting near a boat is definitively the best thing to do if you need to ditch. I have also looked at the US regulations and for small private aircraft, looks like we are not required to carry flotation devices. Just want confirmation. If we plan to go across, I'll have to find the room for some.

Yves

 

Hi Yves, Sorry it took me awhile to check in - VERY busy work week.

 

There is some unknown data - namely the winds aloft and how that changes with altitude, which prevents an exact answer as to what is the critical altitude where you can glide to the shore, and from where at some (not geometric) middle of the lake.  But making some worst case estimates can give errors on the safer side of the error problem.  I will work that up for you tomorrow if I can get a chance.

 

But I consider that an exercise for math students....since when I cross the lakes and its my butt on the line, and I have - I have even been routed right up the middle of Lake Ontario and Lake Erie length wise - at 21,000ft - and I didn't blink since I can glide anywhere I want.  ANd I have crossed lake Huron - at 19,000ft.  Ok, overkill.... but I feel happy if I can glide back to EITHER shore while crossing at all times, despite head winds or tail winds.

[yvesg]

However, as a C-registered plane, you are required to carry floation in accord w/ TC regulations.

Does this apply in USA too? I tought that this would apply only if I fly in Canada.

I checked the minimum crossing distance and I could glide to shore if I fly above 9 thousands assuming no wind...

jlunseth, I do like your comment and, this opens a new perspective....which bring me to the suggestion Bob made early in the thread.... go ditch near a boat!

Where is Erik?

Yves

[yvesg]

Erik, I saved my message before seeing your reply, sorry about that... no rush, I am leaving for Oshkosh only next week.

Yves

[BorealOne]

Hey Yves - yes, I double checked and your C-registered aircraft is always subject to Canadian regulations, in addition to the requirements that may be effect when flying in other nations airspace.

 

On life preservers/life rafts specifically, see CARS 602 Division II 602.57, 602.62 and 602.63. 

[Awful_Charlie]

I'm quite regularly over the English channel, which on the short route is only a bit over 20nm, but Southampton-Jersey is nearer 100nm and Le Harvre-Exeter nearly 150nm.  I tend not to think of the distance to land, but more what the ditching options are - smaller boats are more likely to be able to divert and reach you than large ships (which can't manoever as easily), and ditching has it's own set of skills (land into wind unless there is a large swell, in which case land parallel with the swell, ideally on the top of a crest).  I'll often start a stop watch over a suitable looking rescue possibility, and then when it is behind (and no longer easily visible) if it goes quiet up front, hopefully I have an idea if I'm turning back to it or continuing to one visible in front of me.

 

Altitude give you some more options of course, but if another 1000' puts you into IMC then you might lose any advantage.  Being a bit paranoid, I tend to avoid power changes of any type when out of gliding range, and will stick with the same fuel tank until back in gliding range again (how sad would it be at the mid-point to have the fuel tap come off in your hand?)

 

If the water is cold, a life-vest is only going to make the body easy to pick up   Life rafts, immersion suits all have their place and overheads, but I feel detailed planning may be making life more difficult than you need to - how well do you know the wind gradient, where are rescue services based, what if it is not a total power loss, what if the wind where you are is not as forecast (and have you got a way to readily measure this?) and so on.  I'm not advocating ignoring the risks, but somewhere you have to add a bit of real life!

[yvesg]

Hey Yves - yes, I double checked and your C-registered aircraft is always subject to Canadian regulations, in addition to the requirements that may be effect when flying in other nations airspace.

 

On life preservers/life rafts specifically, see CARS 602 Division II 602.57, 602.62 and 602.63.

I tought that pilot licensing was bound by ICAO but that aircraft equipment would fall on the country specifics regulations.

Thanks for the info.

Yves

[wiguy]

 I've done Lake MI a few times. Yes, I would make room for a life preserver, one for each. Have it handy or even wear it. I fly HIGH, 11,500 or 12,500 for a bit, no OXY.

 

  The lake is narrower North of MKE, find a narrow spot, some islands can help. Most ditichings are intitially survivable, the cold water or drowning is what kills.  There is usually a boat or 3, (frieghters) out near the middle. My plan was to tell ATC & ditch < 1 mile in front, slightly to the side. Check the winds aloft, factor that it. In the Summer there are usually a fair amount of pleasure boats the last 3 miles to shore.

 

  Now we have to back up to the reasons engines quit, don't run out of fuel, or take a questionable plane over water(anywhere really).

 

  I don't advocate flying over large bodies of water, don't even always do it myself. Just think that with planning & thought, risks can be reduced.

[jetdriven]

I would think the point of no return is the point where you would not be within gliding distance anymore, and not necessarily the point where the time required to go forward equals the time to return. It may be way before the halfway point. Not sure yet how to compute it...but you will need to figure out your gliding distance. It will of course vary depending on the wind and direction. Xavion would be helpful in the scenario you're describing.

That's known as a "wet gap" and many corporate jets have this problem as well.  The ETP is strictly to consider a turnback or proceed if having trouble.  If the water really is 61 degrees you better have a nice raft or a survival suit, or else it wont matter if you survive the ditching, you will die of hypothermia.

[Awful_Charlie]

Not looking good for the single crew of an SR22 earlier today   http://www.bbc.co.uk/news/uk-23399158

 

Stay safe out there