Question

Three carts move on a frictionless track with inertias and velocities as shown. The carts collide and stick together after successive collisions.
1) Find loss of mechanical energy when B & C stick together.
2)Find magnitude of impulse experienced by A when it sticks to combined mass (B&C)

Answer 1

1) By conservation of linear momentum,

$m_2{v}_2+m_3{v}_3=(m_2+m_3)v_3\Rightarrow 1\times 1+2(-2)=3\times v_3\therefore v_4=-1㎨$

$LossofM.E={KE}_{Initial}-{KE}_{Final}=\frac{1}{2}{m}_2{v}_2^2+\frac{1}{2}{m}_3{v}_3^2-\frac{1}{2}{(m}_2+m_3)v_4^2=\frac{1}{2}\times 1\times 1^2+\frac{1}{2}\times 2\times {(-2)}^2-\frac{1}{2}\times 3\times {(-1)}^2=3J$

2) $m_1{v}_1+(m_2+m_3)v_4=(m_1+m_2+m_3)v_5\therefore v_5=\frac{2+3(-1)}{5}=-\frac{1}{5}㎧$

$\therefore$ Impulse=change in momentum

$=m_1{v}_1-(m_2+m_3)v_5=2\times 1+5\times (-\frac{1}{5})=1Ns$