Question

Two particles are projected from the same point with same speed $u$ at angles of projection $α$ and $β$ from
horizontal. The maximum heights attained by them are h₁ and h₂ respectively, R is the range for both. If
t₁ and t₂ are their times of flight respectively, then which amongst the option{s) is/are incorrect?

Moderate

Solution

Range becomes equal at complimentary angle. Hence,
$β = 90 ° - α or α + β = 90 ° = π / 2$
(b) $h_{1} = \frac{u^{2} \sin^{2} α}{2 g}$
$\Rightarrow h_{2} = \frac{u^{2} \cos^{2} α}{2 g}$ $(As, β = 90 ° - α)$
$∴ 4 \sqrt{h_{1} h_{2}} = \frac{2 u^{2} \sin α \cos α}{g} = \frac{u^{2} \sin 2 α}{g} = R$
(c) $\frac{t_{1}}{t_{2}} = \frac{(2 u \sin α / g)}{(2 u \cos α / g)} = \tan α$
(d) $\sqrt{\frac{h_{1}}{h_{2}}} = \tan α$

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