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Current Question (ID: 8922)

Question:

$\text{Two cars are moving in the same direction with the same speed 30 km/hr. They are separated by a distance of 5 km. The speed of a car moving in the opposite direction, if it meets these two cars at an interval of 4 minutes, will be:}$

Options:

1. $40 \text{ km/hr}$
2. $45 \text{ km/hr}$
3. $30 \text{ km/hr}$
4. $15 \text{ km/hr}$

Solution:

$\text{Hint: Use the concept of relative velocity.}$ $\text{Step: Find the velocity of the car.}$ $\text{The two cars (say A and B) are moving with the same velocity, the relative velocity of one (say B) with respect to the other}$ $A, \vec{v}_{BA} = \vec{v}_B - \vec{v}_A = v - v = 0$ $\text{So the relative separation between them (= 5 km) always remains the same.}$ $\text{Now if the velocity of car (say C) moving in opposite direction to A and B, is } \vec{v}_C \text{ relative to ground then the velocity of car C relative to A and B will be } \vec{v}_{rel} = \vec{v}_C - \vec{v}$ $\text{But as } \vec{v} \text{ is opposite to } v_C$ $\text{So } v_{rel} = v_C - (-30) = (v_C + 30) \text{ km/hr.}$ $\text{So, the time taken by it to cross the cars A and B}$ $t = \frac{d}{v_{rel}} \Rightarrow \frac{4}{60} = \frac{5}{v_C + 30}$ $\Rightarrow v_C = 45 \text{ km/hr.}$ $\text{Hence, option (2) is the correct answer.}$

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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
}