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Current Question (ID: 8894)

Question:

$\text{The initial velocity of a particle is } u \text{ (at } t = 0\text{) and the acceleration } f \text{ is given by } at\text{. Which of the following relation is valid?}$

Options:

1. $v = u + at^2$
2. $v = u + \frac{at^2}{2}$
3. $v = u + at$
4. $v = u$

Solution:

$\text{If acceleration is variable (depends on time), then,}$ $f = \frac{dv}{dt}$ $dv = f \, dt$ $\text{Given that acceleration } f = at\text{, we integrate:}$ $\int_u^v dv = \int_0^t (at) \, dt$ $\text{Solving the left side: } \int_u^v dv = v - u$ $\text{Solving the right side: } \int_0^t at \, dt = a \int_0^t t \, dt = a \left[\frac{t^2}{2}\right]_0^t = \frac{at^2}{2}$ $\text{Therefore: } v - u = \frac{at^2}{2}$ $v = u + \frac{at^2}{2}$ $\text{Hence, option (2) is the correct answer.}$

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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
}