A ball with a mass of  6 kg is rolling at 25 m/s and elastically collides with a resting ball with a mass of  1 kg. What are the post-collision velocities of the balls?

Apr 12, 2016

${v}_{1}^{'} = \frac{125}{7} \frac{m}{s}$
${v}_{2}^{'} = \frac{300}{7} \frac{m}{s}$

Explanation:

${\vec{P}}_{1} + {\vec{P}}_{2} : \text{ total momentum before collision }$
${\vec{P}}_{1}^{'} + {\vec{P}}_{2}^{'} : \text{ total momentum after collision}$
${\vec{P}}_{1} + {\vec{P}}_{2} = {\vec{P}}_{1}^{'} + {\vec{P}}_{2}^{'} \text{ conservation of momentum}$

${m}_{1} \cdot {v}_{1} + {m}_{2} \cdot {v}_{2} = {m}_{1} \cdot {v}_{1}^{'} + {m}_{2} \cdot {v}_{2}^{'}$
$6 \cdot 25 + 1 \cdot 0 = 6 \cdot {v}_{1}^{'} + 1 \cdot {v}_{2}^{'}$
$150 = 6 \cdot {v}_{1}^{'} + 1 \cdot {v}_{2}^{'} \text{ (1)}$

${v}_{1} + {v}_{1}^{'} = {v}_{2} + {v}_{2}^{'} \text{ (2)}$
$25 + {v}_{1}^{'} = 0 + {v}_{2}^{'}$
${v}_{2}^{'} = 25 + {v}_{1}^{'} \text{ (3)}$

$\text{let's use (1)}$
$150 = 6 \cdot {v}_{1}^{'} + 25 + {v}_{1}^{'}$
$150 - 25 = 7 {v}_{1}^{'}$
$125 = 7 {v}_{1}^{'}$
${v}_{1}^{'} = \frac{125}{7} \frac{m}{s}$

$\text{let's use (3)}$
${v}_{2}^{'} = 25 + \frac{125}{7}$
${v}_{2}^{'} = \frac{175 + 125}{7}$
${v}_{2}^{'} = \frac{300}{7} \frac{m}{s}$