Import Question JSON

Current Question (ID: 9925)

Question:

$\text{A body of mass } m \text{ moving at a certain speed suffers a perfectly inelastic collision with a body of mass } M \text{ at rest. The ratio of the final kinetic energy of the system to the initial kinetic energy will be:}$

Options:

1. $\frac{m}{m+M}$
2. $\frac{M}{m+M}$
3. $\frac{m+M}{m}$
4. $\frac{m+M}{M}$

Solution:

\text{For a perfectly inelastic collision between a mass } m \text{ moving with speed } v \text{ and a mass } M \text{ at rest:} \text{Initial Kinetic Energy: } KE_{\text{initial}} = \frac{1}{2}mv^2 \text{By conservation of momentum, the final velocity of the combined mass } (m + M) \text{ is } V_f = \frac{mv}{m+M}. \text{Final Kinetic Energy: } KE_{\text{final}} = \frac{1}{2}(m + M)V_f^2 = \frac{1}{2}(m + M)\left(\frac{mv}{m+M}\right)^2 = \frac{1}{2}\frac{m^2v^2}{m+M} \text{Ratio of final to initial kinetic energy:} \frac{KE_{\text{final}}}{KE_{\text{initial}}} = \frac{\frac{1}{2}\frac{m^2v^2}{m+M}}{\frac{1}{2}mv^2} = \frac{m}{m+M} \text{The final answer is } \frac{m}{m+M}.

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
}