# A ball with a mass of  5 kg is rolling at 8 m/s and elastically collides with a resting ball with a mass of 2 kg. What are the post-collision velocities of the balls?

Apr 8, 2016

${v}_{1}^{'} = \frac{24}{7} \text{ } \frac{m}{s}$
${v}_{2} = \frac{80}{7} \text{ } \frac{m}{s}$

#### Explanation:

${m}_{1} = 5 \text{ } k g$
${v}_{1} = 8 \text{ } \frac{m}{s}$

${m}_{2} = 2 \text{ } k g$
${v}_{2} = 0$

$\text{momentum before collision:}$
${P}_{b} = {m}_{1} \cdot {v}_{1} + {m}_{2} \cdot {v}_{2}$
${P}_{b} = 5 \cdot 8 + 2 \cdot 0$
${P}_{b} = 40 + 0$
${P}_{b} = 40 \text{ } k g \cdot \frac{m}{s}$

$\text{momentum after collision:}$
${P}_{a} = {m}_{1} \cdot {v}_{1}^{'} + {m}_{2} \cdot {v}_{2}^{'}$
${P}_{a} = 5 \cdot {v}_{1}^{'} + 2 \cdot {v}_{2}^{'}$

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

$40 = 5 \cdot {v}_{1}^{'} + 2 \cdot {v}_{2}^{'} \text{ (1)}$

${m}_{1} \cdot {v}_{1} + {m}_{2} \cdot {v}_{2} = {m}_{1} \cdot {v}_{1}^{'} + {m}_{2} \cdot {v}_{2}^{'} \text{ (3)}$
$\frac{1}{2} {m}_{1} \cdot {v}_{1}^{2} + \frac{1}{2} \cdot {m}_{2} \cdot {v}_{2}^{2} = \frac{1}{2} \cdot {m}_{1} \cdot {v}_{1}^{' 2} + \frac{1}{2} \cdot {m}_{2} \cdot {v}_{2}^{2} ' \text{ (4)}$

$\text{we obtain the equation of "v_1+v_1^'=v_2+v_2^'" ;}$
$\text{if we arrange together the equation (3) and (4)}$

$8 + {v}_{1}^{'} = 0 + {v}_{2}^{'}$
${v}_{2} = 8 + {v}_{1}^{'} \text{ } \left(5\right)$

$\text{now; let's use the equation of (1)}$
$40 = 5 \cdot {v}_{1}^{'} + 2 \cdot \left(8 + {v}_{1}^{'}\right)$
$40 = 5. {v}_{1}^{'} + 16 + 2 \cdot {v}_{1}^{'}$
$40 - 16 = 7 \cdot {v}_{1}^{'}$
$24 = 7 \cdot {v}_{1}^{'}$
${v}_{1}^{'} = \frac{24}{7} \text{ } \frac{m}{s}$

$\text{now;let's use the equation of (5)}$
${v}_{2} = 8 + \frac{24}{7}$

${v}_{2} = \frac{56 + 24}{7}$

${v}_{2} = \frac{80}{7} \text{ } \frac{m}{s}$

$\text{is solution true ?}$
$\cancel{\frac{1}{2}} {m}_{1} \cdot {v}_{1}^{2} + \cancel{\frac{1}{2}} \cdot {m}_{2} \cdot {v}_{2}^{2} = \cancel{\frac{1}{2}} \cdot {m}_{1} \cdot {v}_{1}^{' 2} + \cancel{\frac{1}{2}} \cdot {m}_{2} \cdot {v}_{2}^{2} '$

$5 \cdot {8}^{2} + 0 = 5 \cdot {\left(\frac{24}{7}\right)}^{2} + 2 \cdot {\left(\frac{80}{7}\right)}^{2}$

$5 \cdot 64 = 5 \cdot \frac{576}{49} + 2 \cdot \frac{6400}{49}$

$320 = \frac{2880 + 12800}{49}$

$320 = \frac{15680}{49}$

$\textcolor{g r e e n}{320 = 320 \text{ True}}$