Import Question JSON

Current Question (ID: 19245)

Question:

$\text{Two bodies of the same mass are moving with the same speed, but in different directions in a plane. They have a completely inelastic collision and move together thereafter with a final speed which is half of their initial speed. The angle between the initial velocities of the two bodies (in degree) is:}$

Options:

1. $30^\circ$
2. $60^\circ$
3. $90^\circ$
4. $120^\circ$

Solution:

$\text{Hint: Conserve the linear momentum}$ $\text{Step 1: Find the initial momentum}$ $P_i = \sqrt{(mv)^2 + (mv)^2 + 2(mv)(mv) \cos \theta}$ $= \sqrt{2m^2v^2(1 + \cos \theta)}$ $= mv \sqrt{2(1 + \cos \theta)}$ $\text{Step 2: Find the final momentum}$ $P_f = 2m \left(\frac{v}{2}\right) = mv$ $\text{Step 3: Using conservation of momentum and find the angle between initial velocities.}$ $P_i = P_f$ $\Rightarrow mv \sqrt{2(1 + \cos \theta)} = mv$ $\Rightarrow \sqrt{2(1 + \cos \theta)} = 1$ $\text{Squaring both sides,}$ $2(1 + \cos \theta) = 1$ $\Rightarrow (1 + \cos \theta) = \frac{1}{2}$ $\Rightarrow \cos \theta = -\frac{1}{2}$ $\Rightarrow \theta = 120^\circ$

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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
}