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 heights $h_1$ and $h_2$ respectively, then $R = 4\sqrt{h_1h_2}$.
Reason R: Product of said heights, $h_1h_2 = \left(\dfrac{u^2\sin^2\theta}{2g}\right)\cdot\left(\dfrac{u^2\cos^2\theta}{2g}\right)$.
Answer: (A) Both A and R are true and R is the correct explanation of A.
Equal ranges with the same speed need complementary angles $\theta$ and $90^\circ - \theta$, so $h_1 = \dfrac{u^2\sin^2\theta}{2g}$ and $h_2 = \dfrac{u^2\cos^2\theta}{2g}$; R is true.
$$\sqrt{h_1h_2} = \frac{u^2\sin\theta\cos\theta}{2g}\ \Rightarrow\ 4\sqrt{h_1h_2} = \frac{2u^2\sin\theta\cos\theta}{g} = \frac{u^2\sin2\theta}{g} = R$$
So A is true and follows from R.
Solution by Sreeraj P, M.Sc Physics