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Current Question (ID: 8821)

Question:

$\text{A particle starts from rest with constant acceleration. The ratio of space-average velocity to the time-average velocity is:}$ $\text{where time-average velocity and space-average velocity, respectively, are defined as follows:}$ $\langle v \rangle_{time} = \frac{\int v dt}{\int dt}$ $\langle v \rangle_{space} = \frac{\int v ds}{\int ds}$

Options:

1. $\frac{1}{2}$
2. $\frac{3}{4}$
3. $\frac{4}{3}$
4. $\frac{3}{2}$

Solution:

$\text{Hint: } v = \frac{\int v dt}{\int dt}$ $\text{Step: Find the ratio of space-average velocity to the time-average velocity.}$ $\langle v \rangle_{space} = \frac{\int \sqrt{2gx} dx}{\int dx} = \frac{\sqrt{2g} \int x^{\frac{1}{2}} dx}{\int dx} = \frac{(\sqrt{2g}) 2 x^{\frac{3}{2}}}{3x} = \frac{2\sqrt{2gx}}{3} = \frac{2}{3}v$ $\langle v \rangle_{time} = \frac{\int v dt}{\int dt} = \frac{s}{\Delta t} = \frac{a(\Delta t)^2}{2\Delta t} = \frac{a\Delta t}{2} = \frac{v}{2}$ $\text{From (1) and (2) we get;}$ $\frac{\langle v \rangle_{space}}{\langle v \rangle_{time}} = \frac{4}{3}$ $\text{Therefore, the ratio of space-average velocity to the time-average velocity is } \frac{4}{3}\text{.}$ $\text{Hence, option (3) is the correct answer.}$

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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
}