Import Question JSON

Current Question (ID: 19112)

Question:

$\text{A ball is thrown from point } O \text{ at the base of a staircase. Each step has dimensions of } (0.5 \text{ m} \times 0.5 \text{ m}) (\text{width} \times \text{height}). \text{ What is the minimum initial speed required for the ball to land directly on the } 5^{\text{th}} \text{ step using projectile motion?}$

Options:

1. $5 \sqrt{(\sqrt{2} + 1)} \text{ m/s}$
2. $5 \sqrt{2} \text{ m/s}$
3. $5 (\sqrt{2} + 1) \text{ m/s}$
4. $6 \sqrt{(\sqrt{3} + 1)} \text{ m/s}$

Solution:

$\text{Hint: The equation of the trajectory of a projectile is}$ $y = x \tan \theta - \frac{g x^2}{2 v^2 \cos^2 \theta}.$ $y = x \tan \theta - \frac{g x^2}{2 v^2 \cos^2 \theta}$ $(2.5, 2.5) \text{ must lie on this } \Rightarrow 1 = \tan \theta - \frac{g \times 2.5}{2 v^2 \cos^2 \theta}$ $\Rightarrow \frac{25}{2 v^2 \cos^2 \theta} = \tan \theta - 1$ $\Rightarrow v^2 = \frac{25}{2} \left\{ \frac{1 + \tan^2 \theta}{\tan \theta - 1} \right\}$ $\Rightarrow v_{\min} = 5 \sqrt{\sqrt{2} + 1}$ $[\text{Happens when } \tan \theta = \sqrt{2} + 1]$

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
}