2. A ball of mass $\mathrm{m}_{1}$, sliding on a smooth horizontal surface, collides with a ball of mass $\mathrm{m}_{2}$, which was at rest on the same surface. After the central elastic collision, the balls slide in the same direction, with the speed of the ball of mass $\mathrm{m}_{2}$ being 4 times the speed of the ball of mass $\mathrm{m}_{1}$.
1) Find the ratio of the masses of the balls $\frac{m_{2}}{m_{1}}$.
2) Find the ratio of the speed of the ball $m_{1}$ before the collision to its speed after the collision.
2. CSE: $\frac{m_{1} V_{0}^{2}}{2}=\frac{m_{1} V^{2}}{2}+\frac{m_{2}(4 V)^{2}}{2} ; \quad\left(\frac{V_{0}}{V}\right)^{2}=1+16 \frac{m_{2}}{m_{1}}$

$$
\text { CSI: } m_{1} V_{0}=m_{1} V+m_{2} \cdot 4 V ; \quad \frac{V_{0}}{V}=1+4 \frac{m_{2}}{m_{1}}
$$

$(2) \rightarrow(1):$

$1+8 \frac{m_{2}}{m_{1}}+16\left(\frac{m_{2}}{m_{1}}\right)^{2}=1+16 \frac{m_{2}}{m_{1}}$

$L$

![](https://cdn.mathpix.com/cropped/2024_05_06_556368b0f812d0b7a511g-3.jpg?height=257&width=422&top_left_y=1353&top_left_x=1114)

$2 \frac{m_{2}}{m_{1}}=1 ; \quad \frac{m_{2}}{m_{1}}=\frac{1}{2}$

![](https://cdn.mathpix.com/cropped/2024_05_06_556368b0f812d0b7a511g-3.jpg?height=160&width=377&top_left_y=1359&top_left_x=1548)

$(3) \rightarrow(2): \frac{V_{0}}{V_{1}}=1+4 \cdot \frac{1}{2}=3$