Problem — 2D explosion with three fragments

Problem

A stationary $10\;\text{kg}$ object explodes into three fragments of masses $m_1 = 2$, $m_2 = 3$, $m_3 = 5\;\text{kg}$. Fragment 1 leaves with velocity $\vv{v}_1 = (10;\, 0)\;\text{m/s}$, fragment 2 with $\vv{v}_2 = (0;\, -8)\;\text{m/s}$. Find $\vv{v}_3$ and the energy released in the blast.

Solution

Conservation of $\vv{p}$ (the object is at rest, $\vv{p}_\text{tot} = \vv{0}$ before and after), component by component: $m_1 v_{1x} + m_2 v_{2x} + m_3 v_{3x} = 0 \;\Rightarrow\; 2\cdot 10 + 3\cdot 0 + 5\,v_{3x} = 0 \;\Rightarrow\; v_{3x} = -4\;\text{m/s}$ $m_1 v_{1y} + m_2 v_{2y} + m_3 v_{3y} = 0 \;\Rightarrow\; 2\cdot 0 + 3\cdot(-8) + 5\,v_{3y} = 0 \;\Rightarrow\; v_{3y} = +4{,}8\;\text{m/s}$ $\ev{\vv{v}_3 = (-4;\, 4{,}8)\;\mathrm{m/s}}$ Magnitude: $\lvert v_3\rvert = \sqrt{4^2 + 4{,}8^2} = \sqrt{39{,}04} \approx 6{,}25\;\text{m/s}$.

Energy released in the blast = total kinetic energy of the fragments: $E_\text{blast} = \tfrac{1}{2}m_1 v_1^2 + \tfrac{1}{2}m_2 v_2^2 + \tfrac{1}{2}m_3 v_3^2 = \tfrac{1}{2}\cdot 2\cdot 100 + \tfrac{1}{2}\cdot 3\cdot 64 + \tfrac{1}{2}\cdot 5\cdot 39{,}04$ $E_\text{blast} = 100 + 96 + 97{,}6 \approx 294\;\text{J}$ $\ev{E_\text{blast} \approx 294\;\mathrm{J}}$

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Topics: Quantità di moto e urti Concepts: Esplosione · Conservazione della quantità di moto · Energia cinetica Skills: Conservazione della quantità di moto