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Current Question (ID: 9585)

Question:

$\text{Two boys are standing at the ends } A \text{ and } B \text{ of the ground where } AB = a\text{. The boy at } B \text{ starts running in a direction perpendicular to } AB \text{ with velocity } v_1\text{. The boy at } A \text{ starts running simultaneously with velocity } v \text{ and catches the other boy in a time } t\text{, where } t \text{ is:}$

Options:

1. $\frac{a}{\sqrt{v^2 + v_1^2}}$
2. $\frac{a}{\sqrt{v^2 - v_1^2}}$
3. $\frac{a}{v - v_1}$
4. $\frac{a}{v + v_1}$

Solution:

$\text{Hint:}$ $\text{Step: Find the time when the boy A catches the other boy.}$ $\frac{dx}{dt} = v \cos \theta \text{ ... (1)}$ $\text{Here, } v \sin \theta = v_1$ $\Rightarrow \sin \theta = \frac{v_1}{v}$ $\Rightarrow \cos \theta = \frac{\sqrt{v^2 - v_1^2}}{v}$ $\text{Putting the values in equations we get,}$ $\frac{dx}{dt} = \sqrt{v^2 - v_1^2}$ $\int_a^0 dx = \sqrt{v^2 - v_1^2} \int_0^t dt$ $a = \sqrt{v^2 - v_1^2} \cdot \Delta t$ $\Delta t = \frac{a}{\sqrt{v^2 - v_1^2}}$ $\text{Hence, option (2) is the correct answer.}$

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
}