Import Question JSON

Current Question (ID: 9642)

Question:

$\text{A particle is projected with a velocity } v \text{ such that its range on the horizontal plane is twice the greatest height attained by it. The range of the projectile is: (where } g \text{ is acceleration due to gravity)}$

Options:

1. $\frac{4v^2}{5g}$
2. $\frac{4g}{5v^2}$
3. $\frac{v^2}{g}$
4. $\frac{4v^2}{\sqrt{5g}}$

Solution:

$\text{Hint: } R = \frac{v^2 \sin 2\theta}{g}$ $\text{Step: Find the range of the projectile.}$ $\text{Given;}$ $\text{The range on the horizontal plane is twice the greatest height i.e., } R = 2H$ $\Rightarrow \frac{H}{R} = \frac{\tan \theta}{4}$ $\Rightarrow \cot \theta = \frac{1}{2}$ $\Rightarrow \sin \theta = \frac{2}{\sqrt{5}}, \cos \theta = \frac{1}{\sqrt{5}}$ $\text{The range of the projectile is given by;}$ $R = \frac{2v^2 \sin \theta \cos \theta}{g} = \frac{2v^2 \times 2}{\sqrt{5} \sqrt{5g}} = \frac{4v^2}{5g}$ $\text{Hence, option (1) is the correct answer.}$

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
}