Wednesday, February 25, 2009

Submarine Collision Probability 2

Suppose submarines were spherical particles just wandering round the ocean. Occasionally they would hit the shore its surface or its maximum depth and would bounce off and head away again. Submarines do not move randomly but they do not follow fixed ordered paths either as that makes them easy targets.

The submarine was modelled as a cylinder of length 140 meters and radius 6 meters so it has a volume of 63360 meters cubed. If this is instead seen as a sphere this volume would have a radius of 24.73 meters. Forgive the spherical cow nature of this sum. From a previous calculation you have 25 particles (submarines) of volume 63360 cubic meters each in a space the size of 300 million cubic kilometers. How often will one hit another?

A problem very similar to this is how quickly two gasses will mix. Gas particles fly around very quickly air molecules at about 400 meters per second (which is faster then the speed of sound!) and all that stops cigar smoke instantly filling a room is that the gas molecules hit off each other so much they do not just zip across the room. The calculations below are based on an explanation found here. Maxwell and a few other worked out how often particles should collide and called the distance a particle travels without colliding the Mean free path.

Each particle has a volume of 63360 cubic meters. So 25 particles are 1,584,000 cubic meters. Instead of being in a box these submarine particles are in the atlantic with an operating range in depth of 300 meters. So the 'box' these particles are in is of size 30 000 000 000 000 000 cubic meters. Each particle has 12 million cubic kilometers to itself. so in 12 million cubic kilometers of ocean 63360 meters of it are particle. So 1 part in 190 000 000 000 of the available Atlantic is a submarine.

So how many particles per meter is that? 25*63360 submarines in 30 000 000 000 000 000 meters cubed of water.

Whats area is swept out by a particle (submarine) in a given period of time? I figure submarines wander around at 20kph so that's a sweep volume of pi*d*d*x in a distance x. travelling at 20kph that's pi*24.7*24.7*(20kph*1000m*24hr)=919994045 meters swept per day.

Now along this path how long will it on average travel before hitting another particle?

The mean free path formula is

where n is the number of particles per unit volume. And r is 24.7m. So the length you'd expect to travel between collisions is 1/2710* n.

Actually i think I've found a flaw in my reasoning here. I think the correct answer is about once ever 300 years. Which is still a bit too often isn't it? Ill post the calculations later.

No comments: