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Current Question (ID: 19082)

Question:

$\text{Two particles are projected from the same point with the same speed } u, \text{ but at different angles.}$ $\text{They both cover the same horizontal range } R, \text{ but reach different maximum heights, } h_1 \text{ and } h_2.$ $\text{Which of the following relations is correct?}$

Options:

1. $R^2 = h_1 h_2$
2. $R^2 = 16 h_1 h_2$
3. $R^2 = 4 h_1 h_2$
4. $R^2 = 2 h_1 h_2$

Solution:

$\text{Hint: The range of projectiles is the same for complementary angles.}$ $\text{Step 1: Sketch the trajectories of two particles.}$ $\text{Step 2: Find the maximum height in each case.}$ $\Rightarrow h_1 = \frac{u^2 \sin^2 \theta}{2g}$ $\text{similarly,}$ $h_2 = \frac{u^2 \sin^2 (90^\circ - \theta)}{2g} = \frac{u^2 \cos^2 \theta}{2g}$ $\Rightarrow h_2 = \frac{u^2 \cos^2 \theta}{2g}$ $\text{Therefore;}$ $h_1 h_2 = \frac{u^2 \sin^2 \theta \cos^2 \theta}{4g^2}$ $\Rightarrow 4 h_1 h_2 = \frac{u^2 \sin^2 \theta \cos^2 \theta}{g^2}$ $\text{Step 3: Find the common range.}$ $R = \frac{u^2 \sin 2\theta}{g} = \frac{2u^2 \sin \theta \cos \theta}{g}$ $\Rightarrow R^2 = \frac{4u^4 \sin^2 \theta \cos^2 \theta}{g^2}$ $\Rightarrow R^2 = 16 h_1 h_2.$ $\text{Hence, option (2) is the correct answer.}$

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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
}