Import Question JSON

Current Question (ID: 9913)

Question:

$\text{Block A moves on a smooth surface and collides with block B at rest. The maximum energy stored in the spring will be:}$

Options:

1. $\frac{1}{8}mv^2$
2. $\frac{1}{4}mv^2$
3. $\frac{1}{3}mv^2$
4. $\frac{1}{2}mv^2$

Solution:

\text{Hint: At maximum compression, both blocks start moving with the same velocity.} \text{Step 1: Apply momentum conservation to calculate common velocity.} \text{Initially: Block A has velocity } v \text{ and block B is at rest (velocity = 0)} \text{At maximum compression: Both blocks move with velocity } v' \text{Using conservation of momentum: } mv + 0 = mv' + mv' v' = \frac{v}{2} \text{ (At max compression)} \text{Step 2: Apply energy conservation to calculate the maximum energy stored in the spring.} \text{Initial kinetic energy: } \frac{1}{2}mv^2 \text{Final kinetic energy (at max compression): } 2 \times \frac{1}{2}m(v')^2 = 2 \times \frac{1}{2}m\left(\frac{v}{2}\right)^2 = \frac{1}{4}mv^2 \text{By energy conservation: } \frac{1}{2}mv^2 = 2 \times \frac{1}{2}m(v')^2 + E_{\text{sp}} E_{\text{sp}} = \frac{1}{2}mv^2 - \frac{1}{4}mv^2 = \frac{1}{4}mv^2

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
}