Center of Mass, Momentum & Collision

JEE Main level, single correct, medium

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Question 2 of 130

A body of mass 14 kg initially at rest explodes and breaks into three fragments of masses in the ratio 2:2:32: 2: 3. The two pieces of equal masses fly off perpendicular to each other with a speed of 18 m/s18 \mathrm{~m} / \mathrm{s} each. The velocity of the heavier fragment is ____\_\_\_\_ m/s\mathrm{m} / \mathrm{s}.

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Answer: (D) 12212 \sqrt{2}

From mass ratio 2:2:3 with total mass 14 kg: individual masses are 4 kg, 4 kg, and 6 kg.
Setting coordinate axes along the motion directions of the two equal fragments: fragment 1 has momentum p1⃗=4×18i^=72i^\vec{p_1} = 4 \times 18\hat{i} = 72\hat{i} kg⋅m/s, and fragment 2 has p2⃗=4×18j^=72j^\vec{p_2} = 4 \times 18\hat{j} = 72\hat{j} kg⋅m/s.
By momentum conservation, initially at rest: p1⃗+p2⃗+p3⃗=0\vec{p_1} + \vec{p_2} + \vec{p_3} = 0, so p3⃗=−72i^−72j^\vec{p_3} = -72\hat{i} - 72\hat{j} kg⋅m/s.
Velocity of 6 kg fragment: v3⃗=p3⃗6=−12i^−12j^\vec{v_3} = \frac{\vec{p_3}}{6} = -12\hat{i} - 12\hat{j} m/s.
Speed: ∣v3⃗∣=144+144=288=122|\vec{v_3}| = \sqrt{144 + 144} = \sqrt{288} = 12\sqrt{2} m/s.

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