## dan815 one year ago calculate the probability of collision Suppose there are 2 circles of radius r, placed at the mid points along the lengths of a rectangle like the picture, What is the probability of collision between the circles before either circle hits a boundary?, the Particles can have any direction, and they must keep the direction until collision. Both particles travel at same speed

1. dan815

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2. imqwerty

will we consider gravity ?

3. ganeshie8

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4. ganeshie8

I think the locus of intersection points is the perpendicular bisector of the line segment joining those two bubbles

5. dan815

yeah we need to consider that mid line only i think too, since this is case where the speed is equal

6. dan815

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7. ganeshie8

yeah that angle should be same

8. dan815

now with in this angle space itself we should find the prob and then see what the total prob is

9. dan815

i think we should narrow it down to like how many angle different is possible such that a collision is still possible

10. dan815

**how much angle difference**

11. dan815

sometimes i m surprised at my own english hehe

12. ganeshie8

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13. dan815

it doesnt have to be a perfect collision though, it can be off course as long as they touch

14. ganeshie8

cant we treat the bubbles as point masses

15. ganeshie8

16. dan815

this could be a useful one too like a circle, i think there might be some really neat way to solve this one

17. dan815

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18. imqwerty

ummm....i am little confused but is this the answer- probability=L/[2(W+L)]

19. dan815

there is some modular form of solution for this problem if u limit the angles these circles can travel to 2pi*k/n , where 0<k<n, and k,n are integers then you can consider the cases where boundary collision is taken into account and see when the first collision occurs vs another circle, this problem looks very similiar to modular arithmetic problems

20. dan815

tbh i dont know yet qwerty id have to see your full work

21. dan815

here is one way of approaching it

22. dan815

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23. dan815

now there is a perfect collision for every infinitessimal angle by the proportional gets smaller with some factor

24. dan815

as the perfect collision surface area thins out with the angle from the center

25. dan815

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26. imqwerty

what i did was - like if we have a system like this -|dw:1441176400369:dw| and we are asked that what is the probability that a ball dropped from above falls in the smaller square....then the probability is - (ar. of small square)/(ar. of big square) umm wait i think my answer was not correct but can we follow such an approach?

27. dan815

ya it is definately something simliar to that

28. dan815

the only main thing to consider is how the difference in angle changes

29. dan815

as u move above the straight line

30. dan815

for imperfect collisions

31. ganeshie8

thats really a good idea for simplicity maybe treat bubbles as point masses first define two random variables : $$\theta_1, \theta_2$$

32. dan815

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33. dan815

yah that might actually still be equal if we just use point masses in the end

34. ganeshie8

so the probability for perfection collision is $$0$$ because we don't get any area |dw:1441177185300:dw|