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

Dec 16, 2016

$\text{velocities of objects after collision are:}$
$\text{the red ball's velocity :"v_1^'=6/7" "m/s" at positive direction}$
$\text{the blue ball's velocity :"v_2^'=20/7 " "m/s" at positive direction}$

#### Explanation:

$\text{let the velocities of objects after collision be "v_1' " and } {v}_{2}^{'}$

${P}_{1} ' : \text{ momentum of the red ball after collision}$
${v}_{1}^{'} : \text{velocity of the red ball after collision}$

${P}_{1}^{'} = {m}_{1} \cdot {v}_{1}^{'}$
${P}_{1}^{'} = 5 \cdot {v}_{1}^{'}$

${P}_{2}^{'} : \text{momentum of the blue ball after collision}$
${v}_{2}^{'} : \text{velocity of the blue ball after collision}$

${P}_{2}^{'} = 2 \cdot {v}_{2}^{'}$

$\text{vectorial sum of the momentums after collision :}$
${P}_{a} = {P}_{1}^{'} + {P}_{2}^{'}$
${P}_{a} = 5 {v}_{1}^{'} + 2 {v}_{2}^{'}$

${P}_{b} = {P}_{a} \text{ conservation of momentum}$

$10 = 5 {v}_{1}^{'} + 2 {v}_{2}^{'} \text{ } \left(1\right)$

$\text{the total velocities of the objects must be equal}$

${v}_{1} + {v}_{1}^{'} = {v}_{2} + {v}_{2}^{'}$

$2 + {v}_{1}^{'} = 0 + {v}_{2}^{'}$

$\textcolor{red}{{v}_{2}^{'} = 2 + {v}_{1}^{'}} \text{ } \left(2\right)$

$\text{use (1)}$

$10 = 5 {v}_{1}^{'} + 2 \left(\textcolor{red}{2 + {v}_{1}^{'}}\right)$

$10 = 5 {v}_{1}^{'} + 4 + 2 {v}_{1}^{'}$

$10 = 7 {v}_{1}^{'} + 4$

$10 - 4 = 7 {v}_{1}^{'}$

$6 = 7 {v}_{1}^{'}$

$\textcolor{g r e e n}{{v}_{1}^{'} = \frac{6}{7}} \text{ } \frac{m}{s}$

$\text{use (2)}$

${v}_{2}^{'} = 2 + \textcolor{g r e e n}{\frac{6}{7}}$

${v}_{2}^{'} = \frac{20}{7} \text{ } \frac{m}{s}$