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In some of the physics problems I have done, the solution involves taking a snap shot right before collision and analysing the angles between the bodies. The angles between the bodies are supposedly same as angle between velocity components but I do not understand why it would be generally?

Example:

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A ball with with velocity of 10 meter per second initially collides with two other identical balls whose centres are one a line perpendicular to the initial velocity and are in contact with each other. If the ball is aimed at contact point and the surfaces are frictionless. Find velocity of all three balls after collision.

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Here we take snapshot of how the system looks right before collision pushes everything away, and, then say that half angle in the triangle connecting the sphere is same angle as velocity components (30 degrees in this case). Why is this? As in, why are the angles in the triangle corresponding to angle between velocity components? Is this provable using mathematical arugements?

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The forces between two balls are always perpendicular to the surface, so in direction of the radius, where they touch. That is the reason you know the direction of the acceleration.

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    $\begingroup$ ... and the acceleration determines the subsequent velocity after the collision. $\endgroup$ – garyp Jul 29 '20 at 11:35

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