# A ball with a mass of  5 kg is rolling at 6 m/s and elastically collides with a resting ball with a mass of 3 kg. What are the post-collision velocities of the balls?

Apr 17, 2016

$\textcolor{red}{{v}_{1}^{'} = \frac{3}{2} \frac{m}{s}}$
$\textcolor{red}{{v}_{2}^{'} = \frac{15}{2}}$

#### Explanation:

$\text{momentums before collision}$
${P}_{1} = {m}_{2} \cdot {v}_{1} = 5 \cdot 6 = 30 \text{ } k g \cdot \frac{m}{s}$
${P}_{2} = {m}_{2} \cdot {v}_{2} = 3 \cdot 0 = 0$
$\Sigma {\vec{P}}_{b} = {\vec{P}}_{1} + {\vec{P}}_{2} \text{ total momentum before collision}$

$\Sigma {\vec{P}}_{b} = 30 + 0 = 30 \text{ } k g \cdot \frac{m}{s}$

$\text{total momentum after collision must be "30 " } k g . \frac{m}{s}$

$\text{momentums after collision}$
${\vec{P}}_{1}^{'} = {m}_{1} \cdot {v}_{1}^{'} = 5 \cdot {v}_{1}^{'}$
${\vec{P}}_{2}^{'} = {m}_{2} \cdot {v}_{2}^{'} = 3 \cdot {v}_{2}^{'}$

$\Sigma {P}_{a} = {\vec{P}}_{1}^{'} + {\vec{P}}_{2}^{'}$
$\Sigma {P}_{a} = 5 \cdot {v}_{1}^{'} + 3 \cdot {v}_{2}^{'}$

$\Sigma {\vec{P}}_{b} = \Sigma {\vec{P}}_{a}$

$30 = 5 \cdot {v}_{1}^{'} + 3 \cdot {v}_{2}^{'} \text{ "(1)}$

${v}_{1} + {v}_{1}^{'} = {v}_{2} + {v}_{2}^{'}$
$6 + {v}_{1}^{'} = 0 + {v}_{2}^{'}$
${v}_{2}^{'} = 6 + {v}_{1}^{'} \text{ (2)}$

$\text{use the equation (1)}$

$30 = 5 {v}_{1}^{'} + 3 \left(6 + {v}_{1}^{'}\right)$
$30 = 5 {v}_{1}^{'} + 18 + 3 {v}_{1}^{'}$
$30 - 18 = 8 {v}_{1}^{'}$
$12 = 8 {v}_{1}^{2}$
${v}_{1}^{'} = \frac{12}{8}$
$\textcolor{red}{{v}_{1}^{'} = \frac{3}{2} \frac{m}{s}}$

$\text{use (2)}$
${v}_{2}^{'} = 6 + {v}_{1}^{'}$
${v}_{2}^{'} = 6 + 1 , 5$

$\textcolor{red}{{v}_{2}^{'} = \frac{15}{2}}$