Mathematical probability of collision with a ship?

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BlueSkyNick

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A yachtsman is sailing his 10metre boat at 6 knots on his own with the autohelm in control, when he falls asleep at the chart table. He is approaching a shipping lane at right angles, where the vessels are travelling in a single file at 15knots, 1nm apart. The average beam of the vessels is 25m, and length 100m.

What is the chance of the yacht being hit by the bow of one of the ships? Similarly, what is the chance of the yacht sailing into the side of one of the ships?

Please show all workings.
 
Assuming *everyone* involved is asleep, I would say about 1.35% for being hit by the bow, and 3.38% for sailing into the side. 4.73% chance of an incident.

Double these answers if there's a lane going in the opposite direction, too.

HTH /forums/images/graemlins/wink.gif

Andy
 
not so easy with the simple data you have given us.

A ship has a "domain"; roughly speaking it's the area on the sea surface where that ship IS, and where it CAN be if it makes a manoeuvre. You need to incorporate stopping distance, turning circles, and such things as the bow and stern waves which can sink a small craft without a physical collision taking place.

You also need to know the length of the TSS crossing route to calculate the time of the yacht's exposure in the TSS. In practical terms I have never seen big ships following exactly line astern.

When TSS were introduced (? late 1960s) the Royal Institute of Navigation published many mathematical analyses of such putative collisions in their journal. Worth seeing if any current RIN member on the Forum can help retrieve the info. I threw all my journals away about 5 years ago - five feet of shelf space !
 
Aha! I didn't say it was a TSS /forums/images/graemlins/wink.gif If it was, I don't see why its length is relevant anyway - it could be infinity, the yacht's chances are just the same. Not all the ships are the same dimensions or profile, so their domain as you call it will differ.

I was trying to keep it simple, before introducing new variables.

eg if the yacht is steering 180degs, and the first lane of traffic is steering 225degs. There is then a 10mile gap before reaching a second lane of traffic steering 135degs, does it make any difference? If not, then what does affect his chances?
 
[ QUOTE ]
Please show your workings.

[/ QUOTE ]

How *very* dare you ! /forums/images/graemlins/blush.gif













/forums/images/graemlins/smile.gif

Ships are passing a fixed point at 1 every 240 seconds (60*60/15)
and travelling at 15*0.514 = 7.7 m/s
Yacht is travelling at 6*0.514 = 3.1 m/s

Yacht is in danger of being hit by the bow of the ship for the length of time it takes for the yacht to cross the ships' track - 10m yacht, 3.1 m/s = 3.2 seconds (neglects beam of yacht, add ~0.2 seconds to this if you want).

So yacht is a risk of being hit by bows of ship for 3.2 secs out of 240 when it's crossing the lane = 1.3% (~1.4% if you allow for the beam of the yacht).

The risk of collision exists from the bow of the yacht hitting the nearest side of the ship to the furthest side of the ship hitting the stern of the yacht, so the yacht is in danger for 10m (length of yacht) + 25m (beam of ship - assumed to be to all intents and purposes rectangular) = 35m. Time taken for yacht to cross this 35/3.1 = 11.3 seconds.

(Err - I've changed my mind on this bit....

Collision can be with any part of the ship, which takes 100m/7.7m/s ~13s to cross the yacht's track, so at risk of collision for 13+11.3 = 24.3 s every 240 = 10.1%

Of this 24.3, 3.2s would be bow of ship collision already calculated, so risk of contact with other areas is for 24.3-3.2 = 21.1 seconds every 240, or 8.8%. Half of these will be bow of yacht vs ship, half will be ship vs stern of yacht, so chances of yacht sailing into the ship is half this, or 4.4%.

Now let me sit back while you all point out the string of school boy errors in that!

Andy
 
OK, here goes one attempt:

Probability of hitting the side of a ship.....
A ship side occupies 100m in every 1M of a line the yacht must cross. Therefore (assuming 1M = 1846m which might be slightly in error), probability = 100 / (1*1846) = 0.054, or 5.4%

Probability of being hit by a ship's bow.....
There is some part of the yacht in the path of a ship for the time it takes the yacht to sail 35m (i.e. 25 + 10), namely (35*3600)/(6*1846) = 11.4 secs.
A ship's bow crosses the path of the yacht in the time it takes a ship to travel 1 mile, i.e. (1*3600)/15 = 240 secs.
Therefore probability of getting hit by bow = 11.4/240 = 0.0475, or 4.75%.

So, combined likelihood of disaster is about 10%. An alarm clock is definitely called for!!!
 
Interesting science.

Taking a practical approach, I must have crossed the Channel 20 times in the last two years and haven't had to take avoiding action once. I probably haven't been in serious risk of collision in the last five years. Clearly the risk is not nil but experience suggests it is less than 1 in 20 and possibly as low as less than 1 in 100.

PS: I can't say the same for fishing boats which I have to avoid almost every time I go sailing offshore.
 
But then you probably cross the western part; over here in the east I wish they were as far apart as 1nm! Last two crossings we had to hang around the edge of the TSS like an old lady waiting for a gap. One of those occasions there were 4 or 5 yachts all waiting together for the first to play chicken!
 
The answer has nothing to do with maths. If you fall asleep it is certain that you will wake up to find a ship within 1/2 mile on a collision course, even if the horizon was totally clear 10 minutes ago.
 
[ QUOTE ]
The answer has nothing to do with maths. If you fall asleep it is certain that you will wake up to find a ship within 1/2 mile on a collision course, even if the horizon was totally clear 10 minutes ago.

[/ QUOTE ]

If you're guaranteeing I will wake up before a collision I think I can live with that!
 
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