Question

Two balls are projected making angles of 30 and 45 respectively with the horizontal If both have same
velocity at the highest point of their path, then the ratio of their horizontal range is​

Answer 1

$\blue{\bold{\underline{\underline{Answer:}}}}$

$\green{\tt{\therefore{Range_{1}:Range_{2}=1:\sqrt{3}}}}$

$\orange{\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

$\green{\underline \bold{Given :}} \\ \tt:\implies Angle \: of \: projection( \theta) = 30 \degree \\ \\ \tt:\implies Angle \: of \: projection( \theta_{o} ) = 45 \degree \\ \\ \red{\underline \bold{To \: Find :}} \\ \tt: \implies Ratio \: of \: horizontal \: range =?$

• According to given question :

$\tt \circ \: Let \: velocity \: be \: u_{1} \: and \: u_{2} \\ \\ \bold{At \:maximum\: height : } \\ \tt: \implies u_{1}cos \: \theta = u_{2}cos \: \theta_{o} \\ \\ \tt: \implies \frac{ u_{1} }{ u_{2} } = \frac{cos \: \theta_{o} }{cos \: \theta} \\ \\ \tt: \implies \frac{ u_{1} }{ u_{2} } = \frac{cos \: 45 \degree}{cos \: 30 \degree} \\ \\ \tt: \implies \frac{ u_{1} }{ u_{2} } = \frac{ \frac{1}{\sqrt{2}} }{ \frac{ \sqrt{3} }{2} } \\ \\ \tt: \implies \frac{ u_{1} }{ u_{2} } = \frac{\sqrt{2}}{ \sqrt{3}} \\ \\ \green{\tt: \implies u_{1} = u_{2} \frac{\sqrt{2}}{\sqrt{3} }}$

$\bold{For \: Range : } \\ \tt: \implies \frac{ Range_{1} }{ Range_{2} } = \huge{\frac{ \frac{{ u_{1} }^{2} sin \: 2 \: \theta}{g}}{ \frac{ { u_{2}}^{2} sin \: 2 \theta_{o} }{g} } } \\ \\ \tt: \implies \frac{ Range_{1} }{ Range_{2} } = \huge{\frac{ u_{2}^{2}\times \frac{2}{3} \times sin \:60 \degree }{ u_{2}^{2} \: sin \: 90 \degree}} \\ \\ \tt: \implies \frac{ Range_{1} }{ Range_{2} } = \frac{ \frac{ \frac{2}{3}\times \sqrt{3} }{2}}{ \times 1 } \\ \\ \tt: \implies \frac{ Range_{1} }{ Range_{2} } = \frac{2\sqrt{3}}{2\times 3} \\ \\ \green{\tt: \implies Range_{1} : Range_{2} =1:\sqrt{3}}$