Question

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Answer 1

$\huge\underline\mathfrak{Answer}$

1:1

$\textbf{Step-By-Step-Solution}$

$\textbf{Formula used}$

Horizontal Range

$r = \frac{ {u}^{2} \sin(2) \alpha }{g}$

$\textbf{Given}$

$\textbf{Angle of projection }$

$\alpha (1) = 45 - \alpha$

$\alpha (2) = 45 + \alpha$

$\textbf{Using , Formula}$

$\textbf{Step 1}$

For first motion the eq become

$r = \frac{ {u}^{2} \sin(2 ) \alpha (1)}{g}$

$\alpha (1) = 45 - \alpha$

$\textbf{Eq 1}$

$r = \frac{ {u}^{2} \sin2(45 - \alpha ) }{g}$

$\textbf{Step 2}$

For second motion the eq become

$r = \frac{ {u}^{2} \sin(2 ) \alpha (2)}{g}$

$\alpha (1) = 45 + \alpha$

$\textbf{Eq 2}$

$r = \frac{ {u}^{2} \sin2(45 + \alpha ) }{g}$

$\textbf{Step 3}$

$\textbf{Ratio}$

Divide eq (1) by eq(2)

after cancelled the value of $\textbf{g}$

$\frac{r(1)}{ r(2) } = \frac{ {u}^{2} \sin2(45 - \alpha ) }{ {u}^{2} \sin2(45 + \alpha ) }$

$\frac{r(1)}{ r(2) } = \frac{ \sin2(45 - \alpha ) }{ \sin2(45 + \alpha ) }$

$\frac{r(1)}{ r(2) } = \frac{ (\sin90 - \alpha ) }{ \sin(90 + \alpha ) }$

$\frac{r(1)}{r(2)} = \frac{cos2 (\alpha) }{ \cos2( \alpha ) }$

$\textbf{Hence}$

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Horizontal range ratio= 1:1