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Current Question (ID: 8816)

Question:

$\text{A particle moving in a straight line covers half the distance with a speed of } 3 \text{ m/s. The other half of the distance is covered in two equal time intervals with speeds of } 4.5 \text{ m/s and } 7.5 \text{ m/s respectively. The average speed of the particle during this motion is:}$

Options:

1. $4.0 \text{ m/s}$
2. $5.0 \text{ m/s}$
3. $5.5 \text{ m/s}$
4. $4.8 \text{ m/s}$

Solution:

$\text{Hint: Average speed} = \frac{\text{Total Distance travelled}}{\text{Total time taken}}$ $\text{Step: Find the total time taken by the particle.}$ $\text{If } t_1 \text{ and } t_2 \text{ are the time taken by particle to cover first and second half distance respectively.}$ $t_1 = \frac{x/2}{3} = \frac{x}{6} \text{ .......... (i)}$ $x_1 = 4.5t_2$ $x_2 = 7.5t_2$ $x_1 + x_2 = \frac{x}{2} = 4.5t_2 + 7.5t_2$ $t_2 = \frac{x}{24}$ $\text{Total time } t = t_1 + t_2 = \frac{x}{6} + \frac{x}{24} = \frac{x}{4} \text{ ........ (ii)}$ $\text{Step 2: Find the average speed of the particle.}$ $\text{Average speed} = \frac{\text{Total Distance travelled}}{\text{Total time taken}} = \frac{x}{x/4} = 4 \text{ m/s}$ $\text{So, average speed} = 4 \text{ m/s.}$ $\text{Hence, option (1) is the correct answer.}$

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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
}