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Current Question (ID: 9698)

Question:

$\text{A ship } A \text{ is moving westward with a speed of 10 kmph and a ship } B\text{, 100 km South of } A\text{, is moving northward with a speed of 10 kmph. The time after which the distance between them becomes the shortest is:}$

Options:

1. $0 \text{ h}$
2. $5 \text{ h}$
3. $5\sqrt{2} \text{ h}$
4. $10\sqrt{2} \text{ h}$

Solution:

\text{Hint: Using the relative motion approach } v_{BA} = v_B - v_A \text{Step 1: Find the minimum distance and } v_{BA} \text{B moves at an angle of } 45° \text{ with horizontal w.r.t A.} \text{It is clear from the diagram that the shortest distance between ship A and B is AO.} \text{Here:} \frac{OB}{AB} = \cos 45° = \frac{1}{\sqrt{2}} OB = AB \times \cos 45° = 50\sqrt{2} \times \frac{1}{\sqrt{2}} = 50 \text{ km} \text{Relative velocity:} v_{BA} = \sqrt{v_B^2 + v_A^2} = \sqrt{10^2 + 10^2} = \sqrt{200} = 10\sqrt{2} \text{ km/h} \text{Step 2: Find the time taken to reach the shortest distance.} \text{Time taken for them to reach the shortest distance:} t = \frac{OB}{v_{BA}} = \frac{50}{10\sqrt{2}} = \frac{5}{\sqrt{2}} = \frac{5\sqrt{2}}{2} \text{ h} \text{Therefore, } t = 2.5\sqrt{2} \approx 3.54 \text{ h} \text{Hence, option (2) is the correct answer.}

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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
}