A ball with a mass of  5 kg is rolling at 12 m/s and elastically collides with a resting ball with a mass of 2 kg. What are the post-collision velocities of the balls?

May 22, 2018

The velocities are $= 5.14 m {s}^{-} 1$ and $= 17.14 m {s}^{-} 1$

Explanation:

As the collision is elastic, there is conservation of momentum and conservation of kinetic energy.

${m}_{1} {u}_{1} + {m}_{2} {u}_{2} = {m}_{1} {v}_{1} + {m}_{2} {v}_{2}$

$\frac{1}{2} {m}_{1} {u}_{1}^{2} + \frac{1}{2} {m}_{2} {u}_{2}^{21} / 2 {m}_{1} {v}_{1}^{2} + \frac{1}{2} {m}_{2} {v}_{2}^{2}$

Threfore,

$5 \cdot 12 + 2 \cdot 0 = 5 {v}_{1} + 2 {v}_{2}$

$5 {v}_{1} + 2 {v}_{2} = 60$............................$\left(1\right)$

$\frac{1}{2} \cdot 5 \cdot {12}^{2} + \frac{1}{2} \cdot 2 \cdot {0}^{2} = \frac{1}{2} \cdot 5 {v}_{1}^{2} + \frac{1}{2} \cdot 2 {v}_{2}^{2}$

$5 {v}_{1}^{2} + 2 {v}_{2}^{2} = 720$.........................$\left(2\right)$

Solving for ${v}_{1}$ and ${v}_{2}$ in equations $\left(1\right)$ and $\left(2\right)$

$\left\{\begin{matrix}5 {v}_{1} + 2 {v}_{2} = 60 \\ 5 {v}_{1}^{2} + 2 {v}_{2}^{2} = 720\end{matrix}\right.$

$\iff$, $\left\{\begin{matrix}{v}_{2} = \frac{1}{2} \left(60 - 5 {v}_{1}\right) \\ 5 {v}_{1}^{2} + 2 {v}_{2}^{2} = 720\end{matrix}\right.$

$5 {v}_{1}^{2} + 2 \cdot {\left(\frac{1}{2} \left(60 - 5 {v}_{1}\right)\right)}^{2} = 720$

$10 {v}_{1}^{2} + 3600 - 600 {v}_{1} + 25 {v}_{1}^{2} = 1440$

$35 {v}_{1}^{2} - 600 {v}_{1} + 2160 = 0$

${v}_{1} = \frac{600 \pm \sqrt{{\left(- 600\right)}^{2} - 4 \cdot 35 \cdot 2160}}{2 \cdot 35}$

$= \frac{600 \pm 240}{70}$

Therefore,

${v}_{1} = 12 m {s}^{-} 1$ or ${v}_{1} = 5.14 m {s}^{-} 1$

${v}_{2} = 0 m {s}^{-} 1$ or ${v}_{2} = 17.14 m {s}^{-} 1$