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Current Question (ID: 8893)

Question:

$\text{A body is thrown vertically upwards. If the air resistance is to be taken into account, then the time during which the body rises is:}$

Options:

1. $\text{Equal to the time of fall.}$
2. $\text{Less than the time of fall.}$
3. $\text{Greater than the time of fall.}$
4. $\text{Twice the time of fall.}$

Solution:

\text{Let the initial velocity of the ball be } u. \text{Time of rise: } t_1 = \frac{u}{g+a} \text{ and height reached: } h = \frac{u^2}{2(g+a)} \text{where } a \text{ represents the deceleration due to air resistance.} \text{The time of fall } t_2 \text{ is given by:} \frac{1}{2}(g-a)t_2^2 = \frac{u^2}{2(g+a)} \Rightarrow t_2 = \frac{u}{\sqrt{(g+a)(g-a)}} = \frac{u}{g+a} \cdot \sqrt{\frac{g+a}{g-a}} \therefore t_2 > t_1 \text{ because } \sqrt{\frac{g+a}{g-a}} > 1 \text{Since } \sqrt{\frac{g+a}{g-a}} > 1 \text{ when air resistance is present.} \text{Therefore, the time during which the body rises is less than the time of fall.} \text{This occurs because during upward motion, both gravity and air resistance} \text{act downward, while during downward motion, gravity acts downward} \text{but air resistance acts upward.} \text{Hence, option (2) is the correct answer.}

Import JSON File

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
}