Two balls with same mass and initial velocity, are projected at different angles in such a way that maximum height reached by first ball in 8 times higher than that of the second ball. and are the total flying times of first and second ball, respectivel

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Solution

Q.
Two balls with same mass and initial velocity, are projected at different angles in such a way that maximum height reached by first ball is 8 times higher than that of the second ball. $T_{1}$ and $T_{2}$ are the total flying times of first and second ball, respectively, then the ratio of $T_{1}$ and $T_{2}$ is : [2025]

1 $2 \sqrt{2} : 1$

2 2 : 1

3 $\sqrt{2} : 1$

4 4:1

Ans.
(1)

Given, ${(H)}_{1} = 8 \times {(H)}_{2}$

$H_{1} = \frac{u^{2} \sin^{2} θ_{1}}{2 g} and H_{2} = \frac{u^{2} \sin^{2} θ_{2}}{2 g}$

$\frac{u^{2} \sin^{2} θ_{1}}{2 g} = 8 \times \frac{u^{2} \sin^{2} θ_{2}}{2 g}$

$\sin θ_{1} = 2 \sqrt{2} \sin θ_{2} \Rightarrow \frac{\sin θ_{1}}{\sin θ_{2}} = 2 \sqrt{2}$

$T_{1} = \frac{2 u \sin θ_{1}}{g} and T_{2} = \frac{2 u \sin θ_{2}}{g}$

$\Rightarrow \frac{T_{1}}{T_{2}} = \frac{\sin θ_{1}}{\sin θ_{2}} = 2 \sqrt{2}$

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