# A ball with a mass of  2 kg is rolling at 12 m/s and elastically collides with a resting ball with a mass of  9 kg. What are the post-collision velocities of the balls?

May 9, 2016

${\vec{v}}_{1}^{'} = - 7.64 \frac{m}{s}$
${\vec{v}}_{2}^{'} = 4.36 \frac{m}{s}$

#### Explanation:

$\text{momentums before collision}$
$\text{................................................................................................}$
${m}_{1} = 2 k g$
${\vec{v}}_{1} = 12 \frac{m}{s}$
${\vec{P}}_{1} = {m}_{1} \cdot {v}_{1} \text{ (momentum before collision for the first object)}$
$\vec{P} - 1 = 2 \cdot 12 = 24 k g \cdot \frac{m}{s}$

${m}_{2} = 9 k g$
${\vec{v}}_{2} = 0$
${\vec{P}}_{2} = {m}_{2} \cdot {v}_{2} \text{(momentum before collision for the second object)}$
${\vec{P}}_{2} = 9.0$
${\vec{P}}_{2} = 0$

$\Sigma {\vec{P}}_{b} = {\vec{P}}_{1} + {\vec{P}}_{2} \text{( total momentum before collision)}$
$\Sigma {\vec{P}}_{b} = 24 + 0$
$\Sigma {\vec{P}}_{b} = 24 k g \cdot \frac{m}{s}$

$\text{momentums after collision}$
$\text{................................................................................................}$

${P}_{1}^{'} = {m}_{1} \cdot {\vec{v}}_{1}^{'} \text{ momentum after collision for the first object}$
${P}_{1}^{'} = 2 \cdot {\vec{v}}_{1}^{'}$
${P}_{2}^{'} = {m}_{2} \cdot {\vec{v}}_{2}^{'} \text{ momentum after collision for the second object}$

${P}_{2}^{'} = 9 \cdot {\vec{v}}_{2}^{'}$

$\Sigma {\vec{P}}_{a} = {\vec{P}}_{1}^{'} + {\vec{P}}_{2}^{'}$

Sigma vec P_a=2*vec v_1^'+9*vec v_2^'" (total momentum after collision)"

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

$24 = 2 \cdot {\vec{v}}_{1}^{'} + 9 \cdot {\vec{v}}_{2}^{'} \text{ (1)}$

$S o l u t i o n 1 :$
${\vec{v}}_{1} + {\vec{v}}_{1}^{'} = {\vec{v}}_{2} + {\vec{v}}_{2}^{'}$
$12 + {\vec{v}}_{1}^{'} = 0 + {v}_{2}^{'}$
${v}_{2}^{'} = 12 + {\vec{v}}_{1}^{'}$

$\text{using (1)}$

$24 = 2 \cdot {\vec{v}}_{1}^{'} + 9 \left(12 + {\vec{v}}_{1}^{'}\right)$
$24 = 2 {\vec{v}}_{1}^{'} + 108 + 9 {\vec{v}}_{1}^{'}$
$24 - 108 = 11 {\vec{v}}_{1}^{'}$
$- 84 = 11 {v}_{1}^{'}$

${\vec{v}}_{1}^{'} = - 7.64 \frac{m}{s}$

${\vec{v}}_{2}^{'} = 12 + {\vec{v}}_{1}^{'}$

${\vec{v}}_{2}^{'} = 12 - 7.64$

${\vec{v}}_{2}^{'} = 4.36 \frac{m}{s}$

$\text{solution -2:}$

${v}_{1}^{'} = \frac{2 \cdot {\vec{P}}_{b}}{{m}_{1} + {m}_{2}} - {\vec{v}}_{1}$

${v}_{1}^{'} = \frac{2 \cdot 24}{2 + 9} - 12 \text{ } {\vec{v}}_{1}^{'} = \frac{48}{11} - 12$

${v}_{1}^{'} = - 7.64 \frac{m}{s}$

${\vec{v}}_{2}^{'} = \frac{2 \cdot {\vec{P}}_{b}}{{m}_{1} + {m}_{2}} - {\vec{v}}_{2}$

${\vec{v}}_{2}^{'} = \frac{2 \cdot 24}{2 + 9} - 0$

${\vec{v}}_{2}^{'} = \frac{48}{11}$

${\vec{v}}_{2}^{'} = 4.36 \frac{m}{s}$