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Consider a ball moving at a certain speed. It hits an identical ball at rest. After the collision, both the balls move at equal angles $\theta \ne 90^{0}$ (inelastic collision) with the original line of motion.

After the collision, the components of the velocities perpendicular to the original line of motion are equal because of the conservation of linear momentum. So, after the collision, the balls will cover equal distances y along the perpendicular direction at a time t.

Since they move at equal angles $\theta$ from the original line of motion and cover equal distances along the perpendicular direction, they should also cover equal distances x along the original direction at the time t. So, the components of the velocities along the original direction are also equal.

Thus, the speeds of the balls are equal after an inelastic collision if they move at equal angles from the original line of motion.

I may be wrong in my above argument. If I'm wrong, please correct me.

If I'm correct, I have proven this verbally. Now, I want a mathematical justification. Can someone prove this mathematically?

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  • $\begingroup$ Momentum is also conserved along the line connecting the centers of said objects, plus conservation of energy if the collision is elastic. Have you tried just performing the analysis? $\endgroup$
    – Triatticus
    Commented Aug 21, 2023 at 11:16
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    $\begingroup$ Inelastic colission -> unknown amount of energy converted to heat during colission -> unknown final velocities $\endgroup$ Commented Aug 21, 2023 at 13:50

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If it's an elastic collision, you can include conservation of kinetic energy in your analysis along with momentum conservation.

Let $v_1$ and $v_2$ be the velocities after the collision, and $v_0$ be the velocity of the initial ball before the collision.

Along the original line of motion, the momentum is conserved: $$v_1\cos\theta+v_2\cos\theta=v_0$$ and total kinetic energy is conserved to give: $$v_1^2+v_2^2=v_0^2$$ We have two simultaneous equations for $v_1$ and $v_2$ which we can now solve (perhaps give it a try?)

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  • $\begingroup$ I have edited the question considering the case when the collision is inelastic. Will it still be possible to determine their velocities in the inelastic collision? $\endgroup$ Commented Aug 21, 2023 at 13:27
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    $\begingroup$ Inelastic collisions involve energy being lost to the surroundings (through sound, heat etc), so you can't conserve energy anymore. Without more information about how the energy is lost, I don't think you can generally predict $v_1$ and $v_2$. $\endgroup$
    – Garf
    Commented Aug 21, 2023 at 13:53
  • $\begingroup$ You would need to know the Coefficient of Restitution. Refer to this Wikipedia article, where this term is defined and equations are derived for a collision where the kinetic energy is not conserved: en.wikipedia.org/wiki/Coefficient_of_restitution $\endgroup$ Commented Aug 21, 2023 at 22:47
  • $\begingroup$ @Garf I have edited the question again. We can predict the velocities using the conservation of momentum if they move at equal angles from the original line of motion. Because then their velocities would be equal, which I have verbally justified above in my question. But I still want a mathematical explanation of how their speeds are equal. $\endgroup$ Commented Aug 22, 2023 at 14:23

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