Question

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 $h _{1}$ and $h _{2}$ respectively, then $R =4 \sqrt{ h _{1} h _{2}}$
Reason R: Product of said heights. $h _{1} h _{2}=\left(\frac{ u ^{2} \sin ^{2} \theta}{2 g }\right) \cdot\left(\frac{ u ^{2} \cos ^{2} \theta}{2 g }\right)$
Choose the CORRECT answer :

• A. Both $A$ and $R$ are true and $R$ is the correct explanation of $A$.
• B. Both $A$ and $R$ are true but $R$ is NOT the correct explanation of $A$.
• C. $A$ is true but $R$ is false
• D. $A$ is false but $R$ is true

Answer 1

Answer: (A)

$h _{1}=\frac{ u ^{2} \sin ^{2} \theta_{1}}{2 g } h _{2}=\frac{ u ^{2} \sin ^{2} \theta_{2}}{2 g }$
$h _{1} h _{2}=\frac{ u ^{2} \sin ^{2} \theta_{1}}{2 g } \times \frac{ u ^{2} \sin ^{2} \theta_{2}}{2 g }$
$\theta_{2}=90-\theta_{1}$
$h _{1} h _{2}=\frac{ u ^{2} \sin ^{2} \theta_{1}}{2 g } \cdot \frac{ u ^{2} \cos ^{2} \theta_{1}}{2 g }$
$=\left[\frac{ u ^{2} \sin \theta_{1} \cos \theta_{1}}{2 g }\right]^{2}$
$=\left[\frac{ u ^{2} \sin \theta_{1} \cos \theta_{1}}{2 g } \times \frac{2}{2}\right]^{2}=\frac{ R ^{2}}{16}$
$R =4 \sqrt{ h _{1} h _{2}}$
So R is correct explanation of A