Import Question JSON

Current Question (ID: 9906)

Question:

$\text{A smooth sphere of mass } M \text{, moving with velocity } u \text{, directly collides elastically with another sphere of mass } m \text{ at rest. After the collision, their final velocities are } V \text{ and } v \text{, respectively. The value of } v \text{ is:}$

Options:

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

Solution:

\text{Given a 1D elastic collision:} M_1 = M, \quad u_1 = u M_2 = m, \quad u_2 = 0 V_1 = V, \quad V_2 = v \text{1. Conservation of Momentum: } Mu = MV + mv \text{2. Relative Velocity: } u - 0 = v - V \Rightarrow V = v - u \text{Substitute (2) into (1):} Mu = M(v - u) + mv Mu = Mv - Mu + mv 2Mu = v(M + m) v = \frac{2Mu}{M + m} = \frac{2u}{1 + \frac{m}{M}} \text{The final velocity of mass } m \text{ is: } \boxed{v = \frac{2Mu}{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
}