Import Question JSON

Current Question (ID: 19434)

Question:

$\text{A satellite is revolving in a circular orbit at a height } h \text{ from the earth's surface (radius of earth } R; h \ll R).$ $\text{The minimum increase in its orbital velocity required, so that the satellite could escape from the earth's gravitational field is close to: (Neglect the effect of the atmosphere.)}$

Options:

1. $\sqrt{2gR}$
2. $\sqrt{gR}$
3. $\sqrt{\frac{gR}{2}}$
4. $\sqrt{gR}(\sqrt{2} - 1)$

Solution:

$\text{Hint: } \Delta v = v_e - v_o$ $\text{Step 1: Find the orbital velocity.}$ $v_o = \sqrt{\frac{GM}{R}}$ $\text{Step 2: Find the velocity required to escape.}$ $v_e = \sqrt{\frac{2GM}{R}}$ $\text{Step 3: Find the difference between orbital and escape velocities.}$ $\Delta v = v_e - v_o$ $= (\sqrt{2} - 1) \left[ \sqrt{\frac{GM \times R}{R^2}} \right]$ $= (\sqrt{2} - 1) \sqrt{gR}$

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
}