Question

A stone is thrown at a speed u from an inclined plane whose angle of inclination is q. The maximum range when thrown up the plane is Ru& when thrown down the plane is Rd. When the stone is thrown from ground with same speed its maximum horizontal is RH, then Ru,RH & Rd are in

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Answer 1

Given:

The inclination of plane is $\alpha$. The maximum range when thrown up the plane is R_(u) and when thrown down the plane is R_(d) When the stone is thrown from ground with same speed its maximum horizontal is R_(h)

To find:

Relation between R_(u) , R_(d) and R_(h)

Calculation:

For up the incline , the maximum range for the stone comes as :

$R_{u} = \dfrac{ {u}^{2} }{g { \cos}^{2} ( \alpha )} \{1 - \sin( \alpha ) \}$

For down the incline , the max range comes as :

$R_{d} = \dfrac{ {u}^{2} }{g { \cos}^{2} ( \alpha )} \{1 + \sin( \alpha ) \}$

For horizontal throw, the max range of Projectile is :

$R_{h} = \dfrac{ {u}^{2} }{g}$

Expressing R_(u) and R_(d) in terms of R_(h) and equating them ;

So, the required relationship comes as follows:

$\boxed{ \sf{\dfrac{(R_{u} ){ \cos}^{2} ( \alpha )}{1 - \sin( \alpha ) } = \dfrac{(R_{d} ){ \cos}^{2} ( \alpha )}{1 + \sin( \alpha ) } = R_{h} }}$