Import Question JSON

Current Question (ID: 9675)

Question:

$\text{The raindrops are falling with speed } v \text{ vertically downwards and a man is running horizontal road with speed } u. \text{ The magnitude of the velocity of the raindrops with respect to the man is:}$

Options:

1. $v - u$
2. $v + u$
3. $\sqrt{\frac{v^2 + u^2}{2}}$
4. $\sqrt{v^2 + u^2}$

Solution:

\text{Hint: } (\vec{v})_{R,M} = (\vec{v})_{R,G} - (\vec{v})_{M,G} \text{Step 1: Find the magnitude of the velocity of the raindrops with respect to the man.} \text{Assume velocity for rain and man in vector form.} \text{For the magnitude of the velocity of the raindrops with respect to the man, we need to consider both the vertical and horizontal components of the velocities.} (\vec{v})_{R,G} = -v\hat{j} (\vec{v})_{M,G} = u\hat{i} \text{Step 2: Find the magnitude of the relative velocity } (\vec{v})_{R,M} \text{The relative velocity of the raindrops with respect to the man can be found using:} (\vec{v})_{R,M} = (\vec{v})_{R,G} - (\vec{v})_{M,G} \text{Substituting the known values we get,} (\vec{v})_{R,M} = -v\hat{j} - u\hat{i} \text{The magnitude of the velocity of the raindrops with respect to the man is given by:} |(\vec{v})_{R,M}| = \sqrt{v^2 + u^2} \text{Hence, option (4) 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
}