Import Question JSON

« Back to Question List Edit Question »

Current Question (ID: 9910)

Question:
$\text{Five balls are placed one after another along a straight line as shown in the figure. Initially, all the balls are at rest. Then the second ball is projected with speed } v_0 \text{ towards the third ball. Mark the correct statement(s). (Assume all collisions to be head-on and elastic):}$
Options:
  • 1. $\text{The total number of collisions in the process is 5.}$
  • 2. $\text{The velocity of separation between the first and fifth ball after the last possible collision is } v_0.$
  • 3. $\text{Finally, three balls remain stationary.}$
  • 4. $\text{All of the above are correct.}$
Solution:
\text{Hint: When } m_1 = m_2, v_{1f} = 0 \text{ \& } v_{2f} = u_1 \text{ if } u_{2i} = 0. \text{Step 1: Analyze collision between balls 2 \& 3.} \text{Ball 2 (2 kg) moving with velocity } v_0 \text{ collides with ball 3 (2 kg) at rest.} \text{After collision, ball 2 comes to rest and ball 3 moves with velocity } v_0. \text{Step 2: Find final velocities after the collision between balls 3 \& 4.} \text{Ball 3 (2 kg) with velocity } v_0 \text{ collides with ball 4 (8 kg) at rest.} \text{Using conservation of momentum: } 2v_0 = 2v_1 + 8v_2 \text{ ... (1)} \text{Using conservation of kinetic energy: } \frac{1}{2}(2)v_0^2 = \frac{1}{2}(2)v_1^2 + \frac{1}{2}(8)v_2^2 \text{ ... (2)} \text{From the collision formula: } v_2 = \frac{v_0 - v_1}{4} \text{Solving the equations:} v_0^2 - v_1^2 = 4\left[\frac{v_0 - v_1}{4}\right]^2 (v_0 - v_1)(v_0 + v_1) = \frac{(v_0 - v_1)^2}{4} \text{This gives: } v_1 = -\frac{3v_0}{5} \text{Step 3: Analyze further collisions and draw the diagram} \text{After the collision between balls 3 \& 4:} \text{Ball 3 velocity: } \frac{3v_0}{5} \text{ (leftward)} \text{Ball 4 velocity: } \frac{2v_0}{5} \text{ (rightward)} \text{Ball 4 collides with ball 5, and ball 3 collides with ball

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
}