Given below are two statements. One is labelled as Assertion A and the other is labelled as Reason R.

Assertion A : Two identical balls A and B thrown with same velocity 'u' at two different angles with horizontal attained the same range R. IF A and B reached the maximum height h₁ and h₂ respectively, then $R = 4\sqrt {{h_1}{h_2}}$

Reason R : Product of said heights.

$${h_1}{h_2} = \left( {{{{u^2}{{\sin }^2}\theta } \over {2g}}} \right)\,.\,\left( {{{{u^2}{{\cos }^2}\theta } \over {2g}}} \right)$$

Choose the correct answer :

<p>When two projectiles are thrown with the same initial velocity 'u' but at complementary angles (say, $\theta$ and $(90^\circ - \theta)$) with the horizontal, they attain the same range R. The formula for the range R of a projectile is:</p> <p>$R = \frac{u^2 \sin 2\theta}{g}$</p> <p>For complementary angles, $2 \theta$ and $180^\circ - 2 \theta$ (which simplifies to the same value for the sine function), the ranges are equal.</p> <p>The maximum height $h_1$ for angle $\theta$ is given by:</p> <p>$h_1 = \frac{u^2 \sin^2 \theta}{2g}$</p> <p>And the maximum height $h_2$ for angle $(90^\circ - \theta)$ is given by:</p> <p>$h_2 = \frac{u^2 \cos^2 \theta}{2g}$</p> <p>Now, multiplying these heights:</p> <p>$$ h_1 h_2 = \left( \frac{u^2 \sin^2 \theta}{2g} \right) \cdot \left( \frac{u^2 \cos^2 \theta}{2g} \right)$$</p> <p>This simplifies to:</p> <p>$h_1 h_2 = \frac{u^4 \sin^2 \theta \cos^2 \theta}{4g^2}$</p> <p>Since $\sin^2 \theta \cos^2 \theta = \left( \frac{\sin 2 \theta}{2} \right)^2 = \frac{1}{4} \sin^2 2 \theta$:</p> <p>$h_1 h_2 = \frac{u^4 \sin^2 2 \theta}{16g^2}$</p> <p>Using the range formula $ R = \frac{u^2 \sin 2 \theta}{g} $, we get:</p> <p>$R^2 = \left( \frac{u^2 \sin 2 \theta}{g} \right)^2$</p> <p>Thus, we have:</p> <p>$4h_1 h_2 = \frac{u^4 \sin^2 2 \theta}{4g^2} = \frac{R^2}{4}$</p> <p>This simplifies to:</p> <p>$R = 4\sqrt{h_1 h_2}$</p> <p>Both the assertion and the reason are correct, and the reason correctly explains the assertion.</p> <p>The correct answer is:</p> <p><strong>Option A :</strong> Both A and R are true and R is the correct explanation of A.</p>