Import Question JSON

Current Question (ID: 10318)

Question:

$\text{Two particles of mass, } 2 \text{ kg and } 4 \text{ kg, are projected from the top of a tower simultaneously, such that } 2 \text{ kg of mass is projected with a speed } 20 \text{ m/s at an angle } 30^\circ \text{ above horizontal and } 4 \text{ kg is projected at } 40 \text{ m/s horizontally. The acceleration of the centre of mass of the system of two particles will be:}$

Options:

1. $\frac{g}{2}$
2. $\frac{g}{4}$
3. $g$
4. $2g$

Solution:

\text{The acceleration of the center of mass of a system depends only on the external forces acting on the system, not on the internal forces or the motion of individual particles.} \text{Since both particles are in free fall under gravity, the only external force acting on the system is the gravitational force.} \text{For the center of mass of the system:} \text{Total mass } = m_1 + m_2 = 2 + 4 = 6 \text{ kg} \text{Total external force } = (m_1 + m_2)g = 6g \text{ downward} \text{Therefore, acceleration of center of mass } = \frac{6g}{6} = g \text{The acceleration of the center of mass is } g \text{ downward, regardless of the initial velocities of the individual particles.}

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
}