Import Question JSON

Current Question (ID: 19047)

Question:

$\text{An object travels a distance } x \text{ with a constant speed } v_1, \text{ and then covers another equal distance } x \text{ with a different constant speed } v_2. \text{ The average velocity } v \text{ for the entire journey is given by:}$

Options:

1. $v = \frac{(v_1 + v_2)}{2}$
2. $\frac{1}{v} = \frac{1}{v_1} + \frac{1}{v_2}$
3. $v = \frac{2v_1v_2}{v_1 + v_2}$
4. $v = \frac{(v_1 - v_2)}{2}$

Solution:

$\text{Hint: } v_{\text{avg}} = \frac{\Delta x}{\Delta t}$ $\text{Step 1: Find the time taken by object in each phase.}$ $t_1 = \frac{x}{v_1} \quad \cdots (1)$ $t_2 = \frac{x}{v_2} \quad \cdots (2)$ $\text{Step 2: Find the average velocity } v \text{ for the entire journey.}$ $v_{\text{avg}} = \frac{\text{Total displacement}}{\text{Total time}}$ $\Rightarrow v = \frac{2x}{t_1 + t_2} = \frac{2x}{\frac{x}{v_1} + \frac{x}{v_2}}$ $\Rightarrow v = \frac{2x}{\frac{x}{v_1} + \frac{x}{v_2}} = \frac{2v_1v_2}{v_1 + v_2}$ $\Rightarrow v = \frac{v_1 + v_2}{2}$ $\text{Hence, option (3) is the correct answer.}$

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
}