# Two balls travelling at different speeds collide in two referentials

In a referential R1, one ball B1 travels at 100 m/s and hits and another identitcal ball B2 that travels at 50 m/s in the same direction. Assuming the material in which the balls are made is such that no energy is dissipated in heat and both balls end up travelling at the same speed after the collision, the speed of both should be according to the conservation of energy be about 79 m/s.

$2\cdot \frac{1}{2}mv^{2}=\frac{1}{2}mv_{B1}^{2}+\frac{1}{2}mv_{B2}^{2}$
where $v_{B1}=100$ and $v_{B2}=50$

Now let's move into a the referential R2 that travels at 50 m/s with respect to R1 in the same direction as B1 and B2. In that referential, B2 is initially immobile. It is hit by B1 that travels at 50 m/s in this new referential. After the collision, according to the conservation of energy, the speed of both balls should be 35 m/s.

$2\cdot \frac{1}{2}mv^{2}=\frac{1}{2}mv_{B1}^{2}+\frac{1}{2}mv_{B2}^{2}$
where $v_{B1}=50$ and $v_{B2}=0$

Now, obvisouly, 35 m/s in R2 is 85 m/s in R1, not 79m/s. Where does the discrepancy come from?

the speed of both should be according to the conservation of energy

If both balls have the same speed after the collision, the collision is inelastic, i.e., kinetic energy is not conserved.

If the balls are identical, then conservation of momentum requires that

$$\mathbf v'_1 + \mathbf v'_2 = \mathbf v_1 + \mathbf v_2 = 150 \mathrm{\frac{m}{s}}$$

If the collision is elastic so that the kinetic energy is conserved, then

$$(v'_1)^2 + (v'_2)^2 = (v_1)^2 + (v_2)^2 = 12500\mathrm{\frac{m^2}{s^2}}$$

Solving both equations simultaneously yields

$$v'_1 = 50 \mathrm{\frac{m}{s}}$$

$$v'_2 = 100 \mathrm{\frac{m}{s}}$$

Thus, it cannot be the case that kinetic energy is conserved and both balls have the same speed after the collision.

So, imposing only conservation of momentum and the stipulation that both balls have the same speed after the collision, the speed after the collision is

$$v' = \frac{v_1 + v_2}{2} = 75 \mathrm{\frac{m}{s}}$$

Well, the inconsistency simply comes from the fact that the hypothesis that the two balls stops at the same speed after the collision is not physical, so clearly it won't be invariant by Galilean boost.