Import Question JSON

Current Question (ID: 9667)

Question:

$\text{A particle is projected with a speed } u \text{ at an angle } \theta \text{ to the horizontal. Radius of curvature at highest point of its trajectory is?}$

Options:

1. $\frac{u^2 \cos^2 \theta}{2g}$
2. $\frac{\sqrt{3}u^2 \cos^2 \theta}{2g}$
3. $\frac{u^2 \cos^2 \theta}{g}$
4. $\frac{\sqrt{3}u^2 \cos^2 \theta}{g}$

Solution:

\text{Hint: At the highest point velocity is perpendicular to the acceleration.} \text{Step 1: For the circular motion } a_c = \frac{v^2}{R} \text{Where } v = \text{tangential velocity, } R = \text{radius of curvature} \text{and } a_c = \text{acceleration perpendicular to } v \text{At the highest point of projectile motion:} \text{Here } V = u\cos\theta \text{and } a_c = g \text{So,} a_c = \frac{v^2}{R} \text{Step 2: Find the radius of curvature} R = \frac{(u\cos\theta)^2}{g} R = \frac{u^2 \cos^2 \theta}{g}

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
}