Import Question JSON

Current Question (ID: 19140)

Question:

$\text{A spaceship in space sweeps stationary interplanetary dust. As a result, its mass increases at a rate } \frac{dM(t)}{dt} = bv^2(t), \text{ where } v(t) \text{ is its instantaneous velocity. The instantaneous acceleration of the satellite is:}$

Options:

1. $-\frac{2bv^3}{M(t)}$
2. $\frac{bv^3}{2M(t)}$
3. $-bv^3(t)$
4. $-\frac{bv^3}{M(t)}$

Solution:

$\text{Hint: } F_{\text{thrust}} = v \frac{dm}{dt}$ $\frac{dm(t)}{dt} = bv^2$ $F_{\text{thrust}} = v \frac{dm}{dt}$ $\text{Force on satellite} = -\vec{v} \frac{dm(t)}{dt}$ $M(t) a = -v(bv^2)$ $a = -\frac{bv^3}{M(t)}$

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
}