Import Question JSON

Current Question (ID: 9566)

Question:

$\text{The position of a particle is given by: } \vec{r} = (3.0t\hat{i} - 2.0t^2\hat{j} + 4.0\hat{k}) \text{ m, where t is in seconds and the coefficients have the proper units for r to be in meters. The magnitude and direction of } \vec{v}(t) \text{ at } t = 1.0 \text{ s are:}$

Options:

1. $4 \text{ m/s, } 53° \text{ with x-axis}$
2. $4 \text{ m/s, } 37° \text{ with x-axis}$
3. $5 \text{ m/s, } 53° \text{ with y-axis}$
4. $5 \text{ m/s, } 53° \text{ with x-axis}$

Solution:

\text{Hint: Use } v(t) = \frac{dr(t)}{dt} \text{Step 1: Find } v(t) v(t) = \frac{d}{dt}(3.0t\hat{i} - 2.0t^2\hat{j} + 4.0\hat{k}) = 3.0\hat{i} - 4.0t\hat{j} \text{Step 2: Find velocity at } t = 1.0 \text{ s} \text{At } t = 1.0 \text{ s: } v = 3.0\hat{i} - 4.0\hat{j} \text{Magnitude: } |v| = \sqrt{3^2 + 4^2} = 5.0 \text{ m/s} \text{Direction: } \theta = \arctan\left(\frac{4}{3}\right) \approx 53 \text{ degrees with x-axis} \text{Therefore, option (4) 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
}