Who first discovered that for a two particle elastic collision in 2 dimensions where the mass of the two particles is the same, the angle between the two final velocity vectors equals 90 degrees?
Any information about the history of this result would be most appreciated.
Have you ever played pool or billiards? Is this the way you expect pool balls to behave? Always good to check understanding against physical systems you can observe (and learn from).
Eric Lord thinks : in the frame of the center of inertia of the two colliding mobiles.
Ron Zamir thinks in the frame of the laboratory, and for non-relativistic speeds.
Stam's answer corrected: "The statement's wrong without additional assumptions, so it's rather pointless to inquire about it."
Have you ever played pool or billiards? Is this the way you expect pool balls to behave? Always good to check understanding against physical systems you can observe (and learn from).
@Jacques
Since the lab-frame does coincide with the center-of-mass frame for suitably chosen initial velocities, Ron's claim can't be right.
Well, well, well! The situation described by Ron actually takes place in laboratory frame in which the position of one of two colliding particles is initially fixed. The billiard example is close, but as the bills are also rotating, not only translating, thus the claim is generally false in this case. I have no idea who invented the original idea/example but it is easy to check/prove that final velocity vectors (of material points!) have to be orthogonal.
Great!
Marek found out what Ron had in mind.
I can't but spelling out the simple proof:
We use units in which the mass has the numerical value 1.
All quantities are 2D velocity vectors. All products are scalar products.
(1) v1=0, v2=v (initial condition)
(2) w1+w2=v (momentum conservation)
(3) w1^2+w2^2 = v^2 (energy conservation)
(2) ==> w1^2 + 2 w1w2 + w2^2 = v^2 ==> (using (3)) w1w2 = 0
An interesting case is the central collision: Here w2=0, w1=v and so the orthogonality is trival since the zero-vector is orthogonal to any vector.
I played a bit with sliding coins on a slick table and found that the right angle comes out only very approximately in most cases (the assumption of elastic collision is violated significantly).
Ron, thank you for having brought up this entertaining topic.
Unfortunately I can't contribute anything to the history of the observation.
Congratulations go to Ulrich! For me it is most interesting how Ulrich guessed the hidden assumption, that one of the masses is in rest before collision. Very interesting!!
Best regards, Joachim
PS. Of course I do not expect an answer about the way the assumption has been guesses:) Instead, my congrats are also related to this important step of the scientific communication: the responders should understand each other, even if the formulations of the messages are not completely precise. Therefore I join to Ulrich's thanks for the Ron question.
Well, well, well - again. Ulrich writes about 2D event, yet in reality his calculations are performed in 1D! He ignores the case when both velocity vectors, after collision, acquire non-vanishing components perpendicular (transverse) to initial vector v - what is the essence of the original question. Of course, this cannot happen to point-like masses but for small bodies - yes, it happens, quite frequently. Thus Ulrich's presentation deals with something known as for Newton cradle.
Dear Marek, Ulrich's calculations are performed on dD vectors, there v w is the scalar product in R^d, and v^2 := v v, whenever v and w are vectors in R^d, where d is any natural greater than 0.
Regards, Joachim
@Joachim
it was Marek who pointed out the 'hidden assumption'. He wrote
" The situation described by Ron actually takes place in laboratory frame in which the position of one of two colliding particles is initially fixed."
It would have been even clearer if he had have written
"The situation described by Ron actually takes place in laboratory frame in which one of two colliding particles is initially at rest."
Thank you for your observation that my calculation works in any dimension.
@Marek
I guess you are working hard and could not take the time to read my long note from beginning to end. So you missed my remarks concerning the vector character of the v's ans w's. Notice also Joachims observation on that point.
@ Ulrich
OK, thanks. Indeed, Marek has immersed this important simplifying assumption within his answer. Thus let us be glad that the problem is now clear. Thus only historical remarks should be sought. My access to internet is to slow to perform this during this eve. Hopefully, tomorrow some books should be already found:)
@Ron,
the renowned German enzyclopedia 'Brockhaus' has a short historical paragraph in its entry on collisions in the sense of physics. It mentions Christiaan Huygens and Rene Descartes as early contributors to the understanding of collisions and Isaac Newton as the first who fully understood and correctly formulated the laws of elastic and inelastic collisions. My most recent general physics textbook by Meschede mentions your case but says nothing about history.
Who first discovered that for two equal-particle elastic collisions, with one particle initially at rest, the angle between the two final velocity vectors equals 90 degrees?
At some point this must have been undiscovered, since the cosine of this angle can vary between 0 and (1/8)(v/c)2 for small v/c, where v is the initial velocity of the other particle. It is a nice exercise in relativistic mechanics, which also classifies as classical mechanics, to derive the exact formula.
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