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Current Question (ID: 9904)

Question:

$\text{A body of mass 2 kg moving with a velocity of } (\hat{i} + 2\hat{j} - 3\hat{k}) \text{ m/s collides with another body of mass 3 kg moving with a velocity of } (2\hat{i} + \hat{j} + \hat{k}) \text{ m/s. If they stick together, the velocity in m/s of the composite body will be:}$

Options:

1. $\frac{1}{5}(8\hat{i} + 7\hat{j} - 3\hat{k})$
2. $\frac{1}{5}(-4\hat{i} + \hat{j} - 3\hat{k})$
3. $\frac{1}{5}(8\hat{i} + \hat{j} - \hat{k})$
4. $\frac{1}{5}(-4\hat{i} + 7\hat{j} - 3\hat{k})$

Solution:

$\text{Given:}$ $m_1 = 2 \text{ kg}$ $\vec{v}_1 = (\hat{i} + 2\hat{j} - 3\hat{k}) \text{ m/s}$ $m_2 = 3 \text{ kg}$ $\vec{v}_2 = (2\hat{i} + \hat{j} + \hat{k}) \text{ m/s}$ $\text{For a perfectly inelastic collision (when bodies stick together), we use conservation of linear momentum:}$ $\vec{P}_i = \vec{P}_f$ $m_1\vec{v}_1 + m_2\vec{v}_2 = (m_1 + m_2)\vec{v}$ $\text{Substituting values:}$ $2(\hat{i} + 2\hat{j} - 3\hat{k}) + 3(2\hat{i} + \hat{j} + \hat{k}) = 5\vec{v}$ $\text{Expanding:}$ $2\hat{i} + 4\hat{j} - 6\hat{k} + 6\hat{i} + 3\hat{j} + 3\hat{k} = 5\vec{v}$ $\text{Combining like terms:}$ $8\hat{i} + 7\hat{j} - 3\hat{k} = 5\vec{v}$ $\text{Therefore:}$ $\vec{v} = \frac{8\hat{i} + 7\hat{j} - 3\hat{k}}{5} = \frac{1}{5}(8\hat{i} + 7\hat{j} - 3\hat{k})$

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Expected JSON Format:

{
  "question": "The mass of carbon present in 0.5 mole of $\\mathrm{K}_4[\\mathrm{Fe(CN)}_6]$ is:",
  "options": [
    {
      "id": 1,
      "text": "1.8 g"
    },
    {
      "id": 2,
      "text": "18 g"
    },
    {
      "id": 3,
      "text": "3.6 g"
    },
    {
      "id": 4,
      "text": "36 g"
    }
  ],
  "solution": "\\begin{align}\n&\\text{Hint: Mole concept}\\\\\n&1 \\text{ mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6] = 6 \\text{ moles of carbon atom}\\\\\n&0.5 \\text{ mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6] = 6 \\times 0.5 \\text{ mol} = 3 \\text{ mol}\\\\\n&1 \\text{ mol of carbon} = 12 \\text{ g}\\\\\n&3 \\text{ mol carbon} = 12 \\times 3 = 36 \\text{ g}\\\\\n&\\text{Hence, 36 g mass of carbon present in 0.5 mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6].\n\\end{align}",
  "correct_answer": 4
}