Import Question JSON

Current Question (ID: 9650)

Question:

$\text{A particle is moving along a circle of radius } R \text{ with constant speed } v_0. \text{ What is the magnitude of change in velocity when the particle goes from point } A \text{ to } B \text{ as shown?}$

Options:

1. $2v_0 \sin \frac{\theta}{2}$
2. $v_0 \sin \frac{\theta}{2}$
3. $2v_0 \cos \frac{\theta}{2}$
4. $v_0 \cos \frac{\theta}{2}$

Solution:

\text{Hint: } \Delta v = \sqrt{v_1^2 + v_2^2 + 2v_1v_2 \cos \theta} \text{Step 1: Draw diagram} \text{Step 2: Use formula for resultant} \Delta v = \sqrt{v^2 + v^2 + 2v^2 \cos(\pi - \theta)} \Delta v = \sqrt{2v^2(1 - \cos\theta)} \Delta \vec{v} = \vec{v} - \vec{u} \text{Step 3: Use trigonometric identity} \cos(\pi - \theta) = -\cos\theta \cos\theta = 1 - 2\sin^2 \frac{\theta}{2} 1 - \cos\theta = 2\sin^2 \frac{\theta}{2} \text{Therefore: } \Delta v = 2v_0 \sin \frac{\theta}{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
}