Question

Prove that in the elastic collision of two objects of identical mass, with one being a target initially at rest, the angle between their final velocity vectors is always 90 degrees.

Answer 1

Step 1: draw a diagram. I'd imagine the body initially at rest at the origin; the moving body approaching it along the negative x-axis.

After the collision, by Conservation of Momentum, the x-components of the two bodies momentum must sum to the original momentum; and the y-components must sum to zero.

Answer 2

I think more important here is the conservation of energy, since the conservation of momentum solely cannot really constrain the angle between the final vectors.

Answer 3

Yes, you'll need both. I was just giving a starting point.

Answer 4

Actually, to proof the 90°, I think you need only the conservation of energy.

Answer 5

I'll be impressed if you can write down an argument with only the conservation of energy.

Answer 6

I did this and I don't know what to do after. attached a file

Answer 7

What about \[ \frac{1}{2} m \vec v_1^2 = \frac{1}{2} m \vec v'_1^2 + \frac{1}{2} m \vec v'_2^2 \]
\[\Rightarrow \vec v_1^2 = \vec v'_1^2 + \vec v'_2^2 \] Now think about the vectorial sum and the Pythagorean theorem and it's clear that this holds only if \[ \vec v'_1 \perp \vec v'_2\]

Answer 8

Okay, I see, you were right. Implicitly I was already using the momentum conservation. Sorry.

Answer 9

I was secretly hoping you were right. It's always good to be surprised. But in this case, I think we really do need both.

Answer 10

I don't even need to use any of the trig stuff to prove? So your equation is basically saying that the lines are in a pythygorean relation. The resultant is the initial momentum.