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Q.
Two projectile thrown at $30^{\circ}$ and $45^{\circ}$ with the horizontal respectively, reach the maximum height in same time. The ratio of their initial velocities is
Time taken to reach maximum height
$ t =\frac{ u \sin \theta}{ g } $
$ \therefore \frac{ u _1 \sin \theta_1}{ g }=\frac{ u _2 \sin \theta_2}{ g }$
$ \Rightarrow u _1 \sin 30= u _2 \sin 45$
$ \Rightarrow \frac{ u _1}{ u _2}=\frac{1 / \sqrt{2}}{1 / 2}=\frac{\sqrt{2}}{1}$