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  1. kshitij

    What is the best shape for a soccer goal post?

    I would like to thank you for your patient and insightful responses throughout this thread and the other one :smile: As mentioned earlier this was not any homework question of significance but rather something I casually made up on my own and yet you were committed to help me throughout this...
  2. kshitij

    What is the best shape for a soccer goal post?

    u/gegenpressing91 on reddit does some beautiful illustrations like the one below, Here we clearly see that for wider shots, he did aim to the opposite post so your assumption is right
  3. kshitij

    What is the best shape for a soccer goal post?

    That is, add the assumption that ##d## is uniformly distributed in ##(-D,D)## then it should be fine right?
  4. kshitij

    What is the best shape for a soccer goal post?

    What I wanted to do was to include all the paths of the ball that we allowed in our probability calculation, i.e., all possible straight lines whose perpendicular distance from origin is ##\leq D## But now I see that all possible paths that intersect the semi-circle of radius ##D## would not...
  5. kshitij

    What is the best shape for a soccer goal post?

    the distance of the point from origin will also be the distance of the line from origin as the line is perpendicular to the line joining origin and that point so you don't need to look at ##d## again since you already fixed that while picking the point
  6. kshitij

    What is the best shape for a soccer goal post?

    I was trying to define the equation of the line in parametric form with a given slope and given distance (perpendicular distance) from origin
  7. kshitij

    What is the best shape for a soccer goal post?

    For every point you pick we can define a unique line which is perpendicular to the line joining that point and origin
  8. kshitij

    What is the best shape for a soccer goal post?

    Sorry I meant, perpendicular to the line joining that point and origin
  9. kshitij

    What is the best shape for a soccer goal post?

    Yes, but isn't it correct that for a given point in that semicircular area, there is only one path possible that passes through that point and is perpendicular to the origin
  10. kshitij

    What is the best shape for a soccer goal post?

    I just want to visualise the probability, saying that "this is the probability when the path of the ball is such that ##d## and ##\theta## are uniformly distributed over ##(-D,D)## and ##(0,\pi)## respectively" doesn't give me any hint about what is happening physically. Saying that "this is the...
  11. kshitij

    What is the best shape for a soccer goal post?

    But only ##d## and ##\theta## are variable, what do you mean by 3 degrees of freedom?
  12. kshitij

    What is the best shape for a soccer goal post?

    I think that you assumed that a shot is hit from a point on the arc of a semicircle of radius ##D## but I meant that a shot hit from any point in the area of the semicircle of radius ##D##, because now we see that ##\theta## is uniformly distributed in ##(0,\pi)## and ##d## (displacement form...
  13. kshitij

    What is the best shape for a soccer goal post?

    I didn't get how you did this? Also if that is not how we can think of this probability then what is the physical meaning of the probability that we calculated?
  14. kshitij

    What is the best shape for a soccer goal post?

    Okay, I think now I understand (but still it doesn't feel right) Anyway, let me ask one final thing, for a given post round or square, ##\dfrac{D\pi-2S}{2D\pi}=\dfrac{D\pi-2R}{2D\pi}## is the probability of a shot that is hit within the semicircular area of radius ##D## going in goal assuming...
  15. kshitij

    How to find centroid of a hemisphere using Pappus's centroid theorem?

    Oh yes, I see it now, If we did the first fold horizontally (and the horizontal fold should be in opposite direction to the unfolding otherwise we will end up with what we started with) instead of vertically then the circles wont be linked and thus that is not what is happening in inverting a...
  16. kshitij

    What is the best shape for a soccer goal post?

    Why is the model I'm talking about a subset of your model? for ##R \gt d \gt -R## aren't both the models equivalent?
  17. kshitij

    What is the best shape for a soccer goal post?

    I am thinking with a different model, consider all possible shots that hit the post, now the probability that a shot hits the sector AB is ##\frac14##, right? the whole post is divided into 4 quarters and hitting any one of these quarter is equally likely? correct? Now back to your model, we...
  18. kshitij

    How to find centroid of a hemisphere using Pappus's centroid theorem?

    Well first I learnt that making a cylinder is relatively easier but making a toroid is way too difficult, the paper doesn't want to change its surface's curvature and IIRC there is a theorem like that by Gauss (??). Anyway, to the conclusion, the blue circle are red circle are not linked like a...
  19. kshitij

    How to find centroid of a hemisphere using Pappus's centroid theorem?

    I think that the linkage should be different in both cases but I should check it myself
  20. kshitij

    What is the best shape for a soccer goal post?

    Also, I have another doubt, Say if I set the value of ##D=R## in the round post, then it should mean that now I'm considering only the shots that hit the round post, so now I get the probability of a shot hitting the round post going in as ##P(round)=\dfrac{D\pi-2R}{2D\pi}=0.18## But that...
  21. kshitij

    What is the best shape for a soccer goal post?

    I need a numerical value of ##D## because that will give us a number, i.e., if ##D=R=6cm## then we get 18% chance as calculated in post #124 instead of getting just that the probability in square and round post are equal, we get that probability in square and round post are equal which is equal...
  22. kshitij

    How to find centroid of a hemisphere using Pappus's centroid theorem?

    After stitching it back I cannot believe how the red circle will still be linked with the blue one?
  23. kshitij

    How to find centroid of a hemisphere using Pappus's centroid theorem?

    As I said before that now it makes sense why the circles won't be linked after we stitch it back the other way because we are literally cutting both the circle such that they changed from a circle to a line, but the cutting isn't allowed
  24. kshitij

    How to find centroid of a hemisphere using Pappus's centroid theorem?

    But why is cutting the strip allowed? wouldn't that change the shape we are dealing with?
  25. kshitij

    How to find centroid of a hemisphere using Pappus's centroid theorem?

    I was thinking about this a lot and even though I don't know how it will happen but I think that it should happen, i.e., the red and blue circles shouldn't be linked after inverting the tube inside out. This is because, the red circle which was initially outside the tube, after flipping has to...
  26. kshitij

    How to find centroid of a hemisphere using Pappus's centroid theorem?

    Wait, can we cut it such that we get a rectangle? doesn't that break any rules? is a toroid with single hole isomorphic to rectangle? I agree that with can cut it in both ways that you mentioned, i.e., cutting it such that we get a short fat cylinder and cutting it such that we get a long...
  27. kshitij

    What is the best shape for a soccer goal post?

    Just tell me whether my this conclusion is correct or not?
  28. kshitij

    What is the best shape for a soccer goal post?

    But our selection depends on what value we choose for ##D## (see post #124) and if we chose ##D## to be large enough then it would also include the cases where the ball didn't hit the post. Isn't setting the same constant value of ##D## for the probability calculation of both posts a like to...
  29. kshitij

    How to find centroid of a hemisphere using Pappus's centroid theorem?

    So that means that we cannot increase the length of the hole such that the tube is converted to a strip? Because by this method it does completely makes sense why the red circle passed through the blue one (because we literally cut and stitched the red circle in the process) But yes if that is...
  30. kshitij

    What is the best shape for a soccer goal post?

    Again I can't understand what you mean here, how am I changing the distribution of angles? And why can't this be used to compare the two post shapes? Let me state the whole method as I understand it. Now, I need diagrams and figures to understand pretty much everything so I will now talk w.r.t...
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