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Collisions with 3 spheres

Discussion in 'Mathematics' started by rubikwizard, Mar 6, 2016.

  1. rubikwizard

    rubikwizard New commenter

    I would appreciate any pointers with the question below involving the collision of a sphere with two stationary spheres. I'm okay when there are two, but three has thrown me! Any help appreciated. Many thanks.
    Link to image of question
  2. adamcreen

    adamcreen Occasional commenter

    Note the third sphere collides perpendicular to the line of centres of the other two. This puts it well outside the Edexcel M2 syllabus. If that doesn't matter, then read on!

    At the instant the third sphere collides, the centres form an equilateral triangle. The impulse is transmitted along the sides of this triangle. So the two original spheres each travel off at 60 degrees to their line of centres. Do CLM in the perpendicular direction, as if you were resolving forces. This will tell you the momentum they have perpendicular, the use their direction of travel to work out the velocity, and hence their speed.
  3. rubikwizard

    rubikwizard New commenter

    Thank you so much for your hints. I've now completed the question and got the same answers as the textbook (Bostock & Chandler Further Mechanics). My solution can be seen here and I would appreciate if you could have a quick look to see if I have missed anything.

    I haven't taught mechanics for about 13 years (I teach Computer Science) and took over a class in January after a colleague left. I realise that this question is beyond what I am teaching (AQA M3) but I have a student who is trying to push himself as he has an offer from Cambridge.

    Once again, many thanks for the time you have put into this.
  4. adamcreen

    adamcreen Occasional commenter

    Looks fine to me. I too did oblique collisions at A Level back in 1988 and have all of the B&C books for reference.
  5. Owen134866

    Owen134866 New commenter

    Hey - am not sure what's in AQA M3, as I have always taught Edexcel up to M3, but you might find some of my resources that are shared on here useful! (you'll have to chop out whichever bits are relevant to the course you're teaching though!)

    M1: https://www.tes.com/teaching-resource/maths-mechanics-1-set-of-lesson-powerpoints-6340615
    M2: https://www.tes.com/teaching-resource/maths-mechanics-2-set-of-lesson-powerpoints-6344013
    M3: https://www.tes.com/teaching-resource/maths-mechanics-3-set-of-lesson-powerpoints-11000759
  6. rubikwizard

    rubikwizard New commenter

    Many thanks for your time adamcreen, much appreciated!

    @Owen134866 - Thanks for sharing your resources, they look very detailed. I will be sure to use them in my teaching.
  7. Adishem

    Adishem New commenter

    @rubikwizard - the only addition I can think of to the solution is a comment about the symmetry of the system, and hence the speeds of A and B being the same after the collision. While fairly obvious in this instance, using symmetry to solve problems can be an important technique, and for a student trying to push themselves, this level of formality could be useful in the future.

    Really interesting question, I have a few excellent students in my M3 (OCR) class, I might try to push them if we get some spare time. Cheers!
  8. rubikwizard

    rubikwizard New commenter

    @Adishem - Thanks for the extra point about symmetry. I have noted it and will mention to the student.

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