Import Question JSON

Current Question (ID: 19031)

Question:

$\text{An engine of a train moving with uniform acceleration passes the signal post with velocity } u \text{ and the last compartment passes the same post with velocity } v. \text{ The velocity with which the middle point of the train passes the signal post is:}$

Options:

1. $\sqrt{\frac{v^2 + u^2}{2}}$
2. $\frac{v-u}{2}$
3. $\frac{2}{v+u}$
4. $\sqrt{\frac{2}{v^2 - u^2}}$

Solution:

$\text{Hint: Use the third equation of motion to find acceleration.}$ $\text{Step 1: Find the acceleration.}$ $\text{The distance travelled by the engine would be the length of the train.}$ $v^2 = u^2 + 2aL$ $\Rightarrow a = \frac{v^2 - u^2}{2L}$ $\text{Step 2: Find the velocity with which the middle point of the train passes the signal post.}$ $\text{The distance travelled by the middle point is equal to half the length of the train.}$ $u'^2 = u^2 + 2a \times \frac{L}{2}$ $\Rightarrow u'^2 = u^2 + \frac{v^2 - u^2}{L} \times \frac{L}{2}$ $\Rightarrow u'^2 = u^2 + \frac{v^2 - u^2}{2}$ $\Rightarrow u' = \sqrt{\frac{v^2 + u^2}{2}}$ $\text{Hence, option (1) is the correct answer.}$

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
}