In the laboratory frame of reference, when a moving object collides elastically and obliquely with a stationary object of the same mass, the objects always move off at a right angle. The proof is very straightforward. However, is the opposite also true? (i.e. when two objects collide and move off at a right angle, they must be of the same mass and the collision must be elastic.) Answer to my textbook says yes, but I think that it is also possible for two objects of different mass to make an inelastic collision and move off at a right angle. Why is it not the case? Can anyone provide a proof for this?
I have a spreadsheet which calculates the result of a 2 body collisions in 2D. I found that with equal masses, one starting at rest, and the two separating at 90 degrees, energy was conserved. With masses that are different it is almost always possible to choose an exit velocity (magnitude and direction) for one that will put it at 90 degrees to the other (and not conserve energy). (By the way, in an elastic collision, if you go into a center of mass system, the speed of each mass relative to the center of mass will be the same leaving as it was approaching the CM. Also, the two masses approach the CM along a line and leave along a (probably) different line.)
As @mmesser314 noted, even in a real scenario and by throwing a ball at 45° with respect to the ground you could achieve a collision from which a 90° angle results, although it involves both two drastically different masses---the one of a ball against the one of the Earth---and energy dissipation.
So no, 90° is not a sufficient condition to conclude the collision was elastic and/or between two identical masses.