# Question #6d21b

Jan 31, 2018

The velocities after collision are $= \frac{{v}_{1} \left({m}_{1} - {m}_{2}\right) + 2 {v}_{2} {m}_{2}}{{m}_{2} + {m}_{1}}$ and $\frac{{v}_{2} \left({m}_{2} - {m}_{1}\right) + 2 {m}_{1} {v}_{1}}{{m}_{1} + {m}_{2}}$

#### Explanation:

In an elastic collision, there is conservation of momentum and conservation of kinetic energy

By conservation of momentum,

${m}_{1} {v}_{1} + {m}_{2} {v}_{2} = {m}_{1} {w}_{1} + {m}_{2} {w}_{2}$...................$\left(1\right)$

By the conservation of kinetic energy,

$\frac{1}{2} {m}_{1} {v}_{1}^{2} + \frac{1}{2} {m}_{2} {v}_{2}^{2} = \frac{1}{2} {m}_{1} {w}_{1}^{2} + \frac{1}{2} {m}_{2} {w}_{2}^{2}$

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

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

From $\left(1\right)$

${m}_{1} \left({v}_{1} - {w}_{1}\right) = {m}_{2} \left({w}_{2} - {v}_{2}\right)$......................$\left(3\right)$

From $\left(2\right)$

${m}_{1} \left({v}_{1}^{2} - {w}_{1}^{2}\right) = {m}_{2} \left({w}_{2}^{2} - {v}_{2}^{2}\right)$

${m}_{1} \left({v}_{1} + {w}_{1}\right) \left({v}_{1} - {w}_{1}\right) = {m}_{2} \left({w}_{2} - {v}_{2}\right) \left({w}_{2} + {v}_{2}\right)$.....$\left(4\right)$

From $\left(3\right)$ and $\left(4\right)$

${v}_{1} + {w}_{1} = {v}_{2} + {w}_{2}$

${w}_{1} = {v}_{2} - {v}_{1} + {w}_{2}$

Plugging this value in $\left(1\right)$

${m}_{1} {v}_{1} + {m}_{2} {v}_{2} = {m}_{1} \left({v}_{2} - {v}_{1} + {w}_{2}\right) + {m}_{2} {w}_{2}$

Solving for ${w}_{2}$

${m}_{1} {v}_{1} + {m}_{2} {v}_{2} = {m}_{1} {v}_{2} - {m}_{1} {v}_{1} + {m}_{1} {w}_{2} + {m}_{2} {w}_{2}$

${w}_{2} \left({m}_{1} + {m}_{2}\right) = {v}_{2} \left({m}_{2} - {m}_{1}\right) + 2 {m}_{1} {v}_{1}$

${w}_{2} = \frac{{v}_{2} \left({m}_{2} - {m}_{1}\right) + 2 {m}_{1} {v}_{1}}{{m}_{1} + {m}_{2}}$

Therefore,

${w}_{1} = {v}_{2} - {v}_{1} + \frac{{v}_{2} \left({m}_{2} - {m}_{1}\right) + 2 {m}_{1} {v}_{1}}{{m}_{1} + {m}_{2}}$

$= \frac{\left({v}_{2} - {v}_{1}\right) \left({m}_{1} + {m}_{2}\right) + \left({v}_{2} \left({m}_{2} - {m}_{1}\right) + 2 {m}_{1} {v}_{1}\right)}{{m}_{1} + {m}_{2}}$

$= \frac{{v}_{2} {m}_{1} + {v}_{2} {m}_{2} - {v}_{1} {m}_{1} - {v}_{1} {m}_{2} + {v}_{2} {m}_{2} - {v}_{2} {m}_{1} + 2 {m}_{1} {v}_{1}}{{m}_{1} + {m}_{2}}$

$= \frac{{v}_{1} \left({m}_{1} - {m}_{2}\right) + 2 {v}_{2} {m}_{2}}{{m}_{2} + {m}_{1}}$