JEE Main 2026 — Center of Mass Momentum and Collision Question with Solution

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 $\overrightarrow{p_1}=4\times 18\hat{i}=72\hat{i}$ kg⋅m/s, and fragment 2 has $\overrightarrow{p_2}=4\times 18\hat{j}=72\hat{j}$ kg⋅m/s.
By momentum conservation, initially at rest: $\overrightarrow{p_1}+\overrightarrow{p_2}+\overrightarrow{p_3}=0$, so $\overrightarrow{p_3}=-72\hat{i}-72\hat{j}$ kg⋅m/s.
Velocity of 6 kg fragment: $\overrightarrow{v_3}=\frac{\overrightarrow{p_3}}{6}=-12\hat{i}-12\hat{j}$ m/s.
Speed: $|\overrightarrow{v_3}|=\sqrt{144+144}=\sqrt{288}=12\sqrt{2}$ m/s.