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 m/s each. The velocity of the heavier fragment is ____m/s.

### Related Formula vecp_textinitial = vecp_textfinal$$\vec{p}_{\text{initial}} = \vec{p}_{\text{final}}$$ M_1vecV_1 + M_2vecV_2 + M_3vecV_3 = 0$$M_1\vec{V}_1 + M_2\vec{V}_2 + M_3\vec{V}_3 = 0$$ ### Core Logic

The total mass is 14 \, mathrmkg$14 \, \mathrm{kg}$ and breaks in ratio 2:2:3$2:2:3$. Let the masses be M_1$M_1$, M_2$M_2$, and M_3$M_3$. M_1 = frac27 times 14 = 4 \, mathrmkg$$M_1 = \frac{2}{7} \times 14 = 4 \, \mathrm{kg}$$ M_2 = frac27 times 14 = 4 \, mathrmkg$$M_2 = \frac{2}{7} \times 14 = 4 \, \mathrm{kg}$$ M_3 = frac37 times 14 = 6 \, mathrmkg$$M_3 = \frac{3}{7} \times 14 = 6 \, \mathrm{kg}$$ ### Step 1: Setup Momentum Equations Since M_1$M_1$ and M_2$M_2$ fly off perpendicular to each other, let their velocity vectors be along the x and y axes. Because the total momentum must be zero, we align the fragments opposite to the final 3rd fragment's direction for simplicity, or just use standard axes. Let vecV_1 = -18hati$\vec{V}_1 = -18\hat{i}$ and vecV_2 = -18hatj$\vec{V}_2 = -18\hat{j}$. Note that the solution simplifies the mass ratio directly to 2, 2, 3$2, 2, 3$ as relative masses for the momentum equation: 2(-18hati) + 2(-18hatj) + 3vecV_3 = 0$$2(-18\hat{i}) + 2(-18\hat{j}) + 3\vec{V}_3 = 0$$ ### Step 2: Solve for Velocity vecV_3 = frac36hati + 36hatj3 = 12hati + 12hatj$$\vec{V}_3 = \frac{36\hat{i} + 36\hat{j}}{3} = 12\hat{i} + 12\hat{j}$$ Magnitude of vecV_3$\vec{V}_3$: |vecV_3| = sqrt12^2 + 12^2 = 12sqrt2 \, mathrmm/s$$|\vec{V}_3| = \sqrt{12^2 + 12^2} = 12\sqrt{2} \, \mathrm{m/s}$$ ### Pattern Recognition In a 3-part explosion from rest, the momentum of the third piece must be equal and opposite to the vector sum of the other two pieces. You can use the ratio of masses directly in the momentum conservation equation instead of absolute masses to save time. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Physics: Center of Mass and Collisions