Question

a ball mass 2kgg is mvoing with velocity 5m/sec elastically collide with a ball of mass 5 kg which is moving with velocity 2m/s find the speed of separation.

Answer 1

Total momentum a total energy of the system is conserved in the elastic collisions. We can write following:

$m_1 v_1+m_2 v_2 = m_1 v_1'+m_2 v_2'$

$m_1v_1^2 + m_2 v_2^2 = m_1v_1'^2 + m_2v_2'^2$ (here we have already divided full equation by 2)

$m_1(v_1 - v_1')=m_2(v_2'-v_2)$ (1)

$m_1(v_1+v_1')(v_1-v_1')= m_2(v_2' +v_2)(v_2'-v_2)$ (2)

dividing (1) by (2):

$v_1 +v_1' = v_2'+v_2$ (3)

After solving (1)+(2) and taking into account (3) we have:

$v_1' = -v_2 + 2 \frac{m_1v_1+m_2v_2}{m_1+m_2} = -2+ 2 \frac{10+10}{7} = 3.714$ m/s.

$v_2'= -v_1+ 2 \frac{m_1v_1+m_2v_2}{m_1+m_2} = -5 + 2 \frac{10+10}{7}=0.714$ m/s.

After the collision the balls are moving into opposite directions, so the speed of separation is

$v_1'+v_2'= 3.714 + 0.714 = 4.428$ m/s.

Answer: $4.428$ m/s.