A body of mass 14 kg initially at rest explodes and breaks into three fragments of masses in the rat

Question

A body of mass 14 kg initially at rest explodes and breaks into three fragments of masses in the ratio $2: 2: 3$. The two pieces of equal masses fly off perpendicular to each other with a speed of $18 \mathrm{~m} / \mathrm{s}$ each. The velocity of the heavier fragment is
$\_\_\_\_$ $\mathrm{m} / \mathrm{s}$.

Professor PC Thomas Classes · Answer

The masses of the fragments are $m_1 = 4 	ext{ kg}$, $m_2 = 4 	ext{ kg}$, and $m_3 = 6 	ext{ kg}$. The two equal pieces move perpendicularly with velocity $18 	ext{ m/s}$. Their momentum vectors are $\vec{p_1} = 4(18)\hat{i} = 72\hat{i}$ and $\vec{p_2} = 4(18)\hat{j} = 72\hat{j}$. The resultant momentum of these two is $p_{12} = \sqrt{72^2 + 72^2} = 72\sqrt{2} 	ext{ kg}\cdot	ext{m/s}$. By conservation of momentum, the third piece must have an equal and opposite momentum: $p_3 = p_{12} \implies 6 v_3 = 72\sqrt{2}$. Thus, the velocity of the heavier fragment is $v_3 = 12\sqrt{2} 	ext{ m/s}$.