Monkey Alive

A park ranger is trying to shoot down a monkey on top of a tree with a tranquilizer gun, as illustrated above. Not realizing that the trajectory of the dart will make a parabola, the ranger aims right at the monkey at an angle of \(\theta=30^\circ\) and shoots, which will eventually make the dart end up below the monkey's feet. At the same time, however, the monkey falls freely from where he stood to escape from the shot. If the speed of the dart is such that it can fly all the way to the tree before the monkey falls to the ground, what is the initial speed \(v_0\) for which the monkey gets shot for sure?

Note: The gravitational acceleration is \(10 \text{ m/s}^2.\)

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