Import Question JSON

Current Question (ID: 19221)

Question:

$\text{A particle of mass } m \text{ moving in the } x \text{ direction with speed } 2v \text{ is hit by another particle of mass } 2m \text{ moving in the } y \text{ direction with speed } v. \text{ If the collision is perfectly inelastic, the percentage loss in the energy during the collision is close to:}$

Options:

1. $44\%$
2. $50\%$
3. $56\%$
4. $62\%$

Solution:

$\text{Hint: } \Delta E = \frac{1}{2} \frac{m_1 m_2}{m_1 + m_2} \left[ \vec{v}_2 - \vec{v}_1 \right]^2$ $\therefore \text{Loss of energy}$ $\Delta E = \frac{1}{2} \frac{m_1 m_2}{m_1 + m_2} \left[ \vec{v}_2 - \vec{v}_1 \right]^2$ $= \frac{1}{2} \frac{(m)(2m)}{m + 2m} \left[ 2v \hat{i} - v \hat{j} \right]^2$ $= \frac{5}{3} mv^2$ $\% \text{ Loss} = \frac{\Delta E}{\frac{1}{2} m (2v)^2 + \frac{1}{2} 2mv^2} \times 100$ $= \frac{\frac{5}{3} mv^2}{3 mv^2} \times 100 \approx 56\%$

Import JSON File

Upload a JSON file containing LaTeX/MathJax formatted question, options, and solution.

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
}