Question

Two stones are projected with same velocity $v$ at an angle $\theta \&(90-\theta) .$ If $H$ and $H_{1}$ are the greatest heights in the two paths, what is the relation between $R, H$ and $H_{1} ?$

• A. $ R=4\sqrt{H{{H}_{1}}} $
• B. $ R=\sqrt{H{{H}_{1}}} $
• C. $ R=4\,H{{H}_{1}} $
• D. None of the above

Answer 1

Answer: (A)

Range of projectile
$R =\frac{2 u^{2} \sin \theta \cos \theta}{g} \ldots $(1)
Height $ H=\frac{u^{2} \sin ^{2} \theta}{2 g} \ldots$(2)
$H_{1}=\frac{u^{2} \sin ^{2}(90-\theta)}{2 g}=\frac{u^{2} \cos ^{2} \theta}{2 g}\ldots$(3)
Then, $H H_{1}=\frac{u^{2} \sin ^{2} \theta u^{2} \cos ^{2} \theta}{2 g 2 g}\ldots$(4)
From Eq. (1), we get
$R^{2} =\frac{4 u^{2} \sin ^{2} \theta u^{2} \cos ^{2} \theta \times 4}{2 g \times 2 g} $
$R =\sqrt{16 HH _{1}} $ [from Eq. $(4) ]$
$=4 \sqrt{ HH _{1}}$