Import Question JSON

Current Question (ID: 19460)

Question:

$\text{Two satellites } S_1 \text{ and } S_2 \text{ are revolving in circular orbits around a planet with radius } R_1 = 3200 \text{ km and } R_2 = 800 \text{ km respectively.}$ $\text{The ratio of the speed of the satellite } S_1 \text{ to the speed of the satellite } S_2 \text{ in their respective orbits would be:}$

Options:

1. $\frac{1}{2}$
2. $\frac{1}{3}$
3. $\frac{1}{4}$
4. $\frac{2}{3}$

Solution:

$\text{Hint: } v = \sqrt{\frac{GM}{r}}$ $\text{Step: Find the ratio of the speeds of the satellites.}$ $\text{The orbital speed of a satellite revolving in a circular orbit is given by:}$ $\Rightarrow v = \sqrt{\frac{GM}{r}}$ $\text{The ratio of the speeds of the satellites is given by:}$ $\Rightarrow \frac{v_1}{v_2} = \sqrt{\frac{r_2}{r_1}}$ $\text{Substitute the known values we get:}$ $\Rightarrow \frac{v_1}{v_2} = \sqrt{\frac{800}{3200}} = \frac{1}{2}$ $\text{Hence, option (1) is the correct answer.}$

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
}