# How to mathematically prove the balls move at equal speeds after an inelastic collision?

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?

• 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? Commented Aug 21, 2023 at 11:16
• Inelastic colission -> unknown amount of energy converted to heat during colission -> unknown final velocities Commented Aug 21, 2023 at 13:50

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?)
• 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 $v_2$.