Two objects are projected with same velocity however at different angles and with the horizontal. If , the ratio of horizontal range of the first object to the second object will be [2023]

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Solution

Q.
Two objects are projected with same velocity $' u'$ however at different angles $α$ and $β$ with the horizontal. If $α + β = 90 °$ , the ratio of horizontal range of the first object to the second object will be [2023]

1 1 : 2

2 4 : 1

3 2 : 1

4 1 : 1

Ans.
(4)

$Range = \frac{u^{2} \sin 2 θ}{g}$

$Range for projection angle$ $'' α''$

$R_{1} = \frac{u^{2} \sin 2 α}{g}$

$Range for projection angle$ $'' β''$

$R_{2} = \frac{u^{2} \sin 2 β}{g}$

$α + β = 90 ° (given) \Rightarrow β = 90 ° - α$

$R_{2} = \frac{u^{2} \sin 2 (90 ° - α)}{g}$

$R_{2} = \frac{u^{2} \sin (180 ° - 2 α)}{g}$

$R_{2} = \frac{u^{2} \sin 2 α}{g}$

$\Rightarrow \frac{R_{1}}{R_{2}} = \frac{(\frac{u^{2} \sin 2 α}{g})}{(\frac{u^{2} \sin 2 α}{g})} = \frac{1}{1}$

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