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I'm having problems proving that for an elastic collision between 2 bodies of equal mass that the objects are separated by 90 degrees after the

I'm having problems proving that for an elastic collision between 2 bodies of equal mass that the objects are separated by 90 degrees after the collision. I am attaching a photo of the other equations, I really am stuck on just the last part, if I can prove the 90 degree separation then I can prove the other equations fully. This is for a collision of two objects with equal masses, the second object is at rest initially, and the collision is fully elastic.

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